Comparing temperaments

Images of tem­pered scales cre­at­ed by the Bol Processor

The fol­low­ing are Bol Processor + Csound inter­pre­ta­tions of Bach’s Prelude 1 in C major (BWV 846) using scales con­struct­ed with mean­tone tem­pera­ments (Asselin 2000). Names and tun­ing pro­ce­dures fol­low Asselin’s instruc­tions (pages 67-126). 

The con­struc­tion of these scales is explained on page Microtonality.

Start lis­ten­ing to the piece in equal tem­pera­ment which is the most com­mon tun­ing of instru­ments in mod­ern times:

Equal tem­pera­ment (p. 123) ➡ Image

The fol­low­ing are tra­di­tion­al mean­tone tem­pera­ments, each of which has been designed at a par­tic­u­lar peri­od in response to the con­straints of musi­cal reper­toires en vogue (Asselin 2000 p. 139-180).

H.A. Kellner’s BACH (p. 101) ➡ Image
Barka in 1786 (p. 106) ➡ Image
Bethisy in 1764 (p. 121) ➡ Image
Chaumont in 1696 (p. 109) ➡ Image
Corrette in 1753 (p. 111) ➡ Image
D’Alambert-Rousseau 1752-1767 (p. 119) ➡ Image
Kirnberger II in 1771 (p. 90) ➡ Image
Kirnberger III in 1779 (p. 106) ➡ Image
Marpurg in 1756 (p. 117) ➡ Image
Pure minor thirds in 16th cen­tu­ry (p. 82) ➡ Image
Rameau en do in 1726 (p. 106) ➡ Image
Sauveur in 1701 (p. 80) ➡ Image
Tartini-Vallotti in mid. 18th cen­tu­ry (p. 104) ➡ Image
Werckmeister III in 1691 (p. 194) ➡ Image
Werckmeister IV in 1691 (p. 96) ➡ Image
Werckmeister V in 1691 (p. 199) ➡ Image
Zarlino in 1558 (p. 85) ➡ Image

The last exam­ple is Zarlino’s mean­tone tem­pera­ment which should not be con­fused with the pop­u­lar Zarlino’s “nat­ur­al scale”, an instance of just into­na­tion:

Zarlino’s “nat­ur­al scale” ➡ Image

Discussion

Comparing tem­pera­ments on a sin­gle piece is a very lim­it­ed exer­cise aimed at high­light­ing dif­fer­ences in the var­i­ous pro­pos­als and their ade­qua­cy to cre­ate a pleas­ant effect when lis­ten­ing to this par­tic­u­lar piece.

J.S. Bach’s dis­ci­ple Johann Kirnberg (1721-1723) - (source)

In real­i­ty, Bach’s Well-Tempered Clavier (BWV 846–893) is a col­lec­tion of two sets of pre­ludes and fugues in all 24 major and minor keys. To assess the valid­i­ty of a tun­ing scheme it would there­fore be nec­es­sary to lis­ten to all pieces. Fortunately, there are clues to an opti­mal choice: Friedrich Wilhelm Marpurg received infor­ma­tion from Bach’s sons and pupils and Johann Kirnberger, one of those pupils, designed a tun­ing (Kirnberger II) which he claimed to rep­re­sent his mas­ter’s idea of “well-tempered”.

Chapter VIII of Pierre-Yves Asselin’s book (2000 p. 139-180) con­tains exam­ples of musi­cal works high­light­ing the rel­e­vance of spe­cif­ic tem­pera­ments. Given that the scores of many (if not all) Baroque and clas­si­cal mas­ter­pieces are avail­able in dig­i­tal for­mat MusicXML, we may use Bol Processor’s Importing MusicXML scores to transcode them and play these excerpts with the sug­gest­ed temperaments.

Musicians inter­est­ed in con­tin­u­ing this research may use Bol Processor BP3’s beta ver­sion to process musi­cal works and cre­ate new tun­ing pro­ce­dures. Follow instruc­tions on page Bol Processor ‘BP3’ and its PHP inter­face to install BP3 and learn its basic oper­a­tion. Download and install Csound from its dis­tri­b­u­tion page.

References

Asselin, P.-Y. Musique et tem­péra­ment. Paris, 1985, repub­lished in 2000: Jobert. Soon avail­able in English.

Importing MusicXML scores

MusicXML is a very pop­u­lar XML-based file for­mat for rep­re­sent­ing Western musi­cal nota­tion. It is designed for the inter­change of scores between scorewrit­ers and oth­er musi­cal devices.

Inside a MusicXML file…

A MusicXML file con­tains all the infor­ma­tion required to dis­play a musi­cal score in com­mon Western music nota­tion. It is also struc­tured in such a way that data can be extract­ed and processed by a sound device to “play” the musi­cal score. The ren­der­ing is pure­ly mechan­i­cal, based on metro­nom­ic tim­ing and lack­ing con­trol over vol­ume, veloc­i­ties, strokes etc. which are not pre­cise­ly dis­played on the score. As such, it may be used as a tool for check­ing the rep­re­sen­ta­tion of a musi­cal work, or an as a teach­ing assis­tant for deci­pher­ing scores.

Importing musi­cal scores from musi­cal archives to Bol Processor makes it pos­si­ble to use them (or frag­ment of them) in gram­mars pro­duc­ing vari­a­tions, for exam­ple Mozart’s musi­cal dice game. Owing to the Csound inter­face, these musi­cal works may also be played back with spe­cif­ic tun­ings as explained on page Microtonality. The lat­ter was an incen­tive for imple­ment­ing MusicXML con­ver­sion, mak­ing it pos­si­ble to try many pieces of the Baroque and clas­si­cal reper­toires against the diver­si­ty of mean­tone tem­pera­ments doc­u­ment­ed by historians.

The MusicXML to Bol Processor con­vert­er is ful­ly oper­a­tional on the PHP inter­face of BP3. Follow instruc­tions on page Bol Processor ‘BP3’ and its PHP inter­face to install BP3 and learn its basic oper­a­tion. Download and install Csound from its dis­tri­b­u­tion page.

Bol Processor’s data format

The Bol Processor has its own data for­mat for rep­re­sent­ing musi­cal items aimed at pro­duc­ing sound via its MIDI or Csound inter­face. This for­mat is dis­played and saved as pure text.

The syn­tax of Bol Processor data is based on poly­met­ric struc­tures. A few ele­men­tary exam­ples will clar­i­fy this term:

  • {A4 B4 C5} is a sequence of three notes “A4”, “B4”, “C5” played at the metro­nom­ic tempo
  • {A4, C5, E5, A5} is a A minor chord
  • {la3, do4, mi4, la4} is the same chord in Italian/Spanish/French convention
  • {dha4, sa5, ga5, dha5} is the same chord in Indian convention
  • {C4 G4 E4, F3 C4} is a two-level struc­ture call­ing for the jux­ta­po­si­tion and time align­ment of sequences “C4 G4 E4” and “F3 C4”, which yields a polyrhyth­mic struc­ture that may be expand­ed as {C4_ G4_ E4_, F3__ C4__} in which ‘_’ are pro­lon­ga­tions of the pre­ced­ing notes.
  • {5, A4 B4 C5} is sequence “A4 B4 C5” played over 5 beats. Durations are mul­ti­plied by 5/3.
  • {7/16, F3 C4} is sequence “F3 C4” played over 7/16 beats. The dura­tion of each note is mul­ti­plied by 7/16/2 = 7/32.

Unlike score rep­re­sen­ta­tion mod­els, poly­met­ric struc­tures are recur­sive with no lim­it in their com­plex­i­ty (except the machine). A few com­plex struc­tures are dis­cussed on page Harm Visser’s exam­ples.

Why do we need to import scores?

Bol Processor’s data for­mat is alto­geth­er com­pact, com­pu­ta­tion­al and com­pre­hen­si­ble by humans. However its com­pact­ness makes it dif­fi­cult to enter com­plex struc­tures. Rather, these struc­tures are pro­duced by grammars…

Even a gram­mar able to cre­ate pieces of tonal music may need some mate­r­i­al extract­ed from exist­ing musi­cal works. Till now (in Bol Processor BP1 and BP2) it was pos­si­ble to map the com­put­er key­board to arbi­trary signs rep­re­sent­ing drum strokes (see the ini­tial project) or notes in com­mon music nota­tion (three dif­fer­ent con­ven­tions) and to import Csound scores or MIDI files to sound-objects.

Things get com­plex when the mate­r­i­al is poly­phon­ic tonal music. Since this mate­r­i­al exists on scores in Western music nota­tion, and these scores have been dig­i­tized to inter­change for­mats such as MusicXML, a import pro­ce­dure cap­tur­ing the whole com­plex­i­ty of the score is a great asset. Mozart’s Musical dice game is a good exam­ple of this necessity.

In prac­tice you could pick up and rework frag­ments of the very large musi­cal reper­toire shared in XML for­mat, or cre­ate your own mate­r­i­al using a score edi­tor such as Werner Schweer’s MuseScore — a public-domain pro­gram work­ing with Linux, Mac and Windows. MuseScore rec­og­nizes many input/output for­mats and it is able to cap­ture music via MIDI or Open Sound Control.

Importing and converting a MusicXML score

A few public-domain MusicXML scores are found in the “xml­sam­ples” of our sam­ple set bp3-ctests-main.zip shared on GitHub. Most of them are frag­ments used for illus­trat­ing the for­mat. We start with a very short frag­ment of “MozartPianoSonata.musicxml” for which the graph­ic score is also available:

Mozart’s piano sonata, an excerpt in com­mon Western music notation

First cre­ate a Data file named for instance “-da.musicXML”. Default set­tings will be suf­fi­cient for this exam­ple, but a “-se.musicXML” file may be declared on the data win­dow, with the result that you will be prompt­ed to cre­ate it. Keep default set­tings as they include graph­ic display.

To import the MusicXML file, click the Choose File but­ton on top of the edit­ing form, select the file and click IMPORT.

The machine will dis­play the list of “parts” con­tained in the score. Each part may be assigned to an instru­ment, includ­ing human voic­es. This score con­tains a unique part to be played on an Acoustic Grand Piano which would be ren­dered by chan­nel 1 of a MIDI device. This chan­nel infor­ma­tion will be put in the Bol Processor score and may be lat­er mapped to Csound instruments.

Clicking CONVERT THEM (or it) is the only things that remains to be done!

This will cre­ate the fol­low­ing Bol Processor data:

// MusicXML file ‘MozartPianoSonata.musicxml’ con­vert­ed
// Score part ‘P1’: instru­ment = Acoustic Grand Piano — MIDI chan­nel 1

-se.musicXML

{_tempo(2) _chan(1){2,{2,C#6},{C#5,E5,A5}-,{1/4,A2 C#3 E3}{1/4,A3}{3/2,A3 A3 A3}}} {_tempo(2) _chan(1){2,{2,D6 C#6 B5 C#6 D6 C#6 B5 C#6},{1/4,A2 C#3 E3}{1/4,A3}{3/2,A3 A3 A3}}}{_tempo(2) _chan(1){2,{2,F#5,A5,D6},{1/4,D2 F#2 A2}{1/4,D3}{3/2,D3 D3 D3}}}{_tempo(2) _chan(1){2,{1/8,D6}{3/8,E5,A5,C#6}{1/8,D6}{3/8,E5,A5,C#6}{1/8,D6}{3/8,E5,A5,C#6}{1/8,D6}{3/8,E5,A5,C#6},{1/4,A2 C#3 E3}{1/4,A3}{3/2,A3 A3 A3}}}{_tempo(2) _chan(1){2,{3/2,B5}{1/2,E6},{2,E5,G#5},{1/4,E2 G#2 B2}{1/4,E3}{3/2,E3 E3 E3}}}

Imported scores can be played, expand­ed, explod­ed and imploded

Indeed this looks uneasy to read, but remem­ber that a lay per­son would not even make sense of the score in com­mon Western music nota­tion! Fortunately, a PLAY but­ton is now avail­able to lis­ten to the piece. By default, it is saved as a MIDI file which then can be inter­pret­ed by a MIDI soft syn­the­siz­er such as PianoTeq:

Mozart’s piano sonata, an excerpt played by Bol Processor with PianoTeq

The same process can be invoked in the Csound envi­ron­ment. If Csound is installed and respon­sive, select­ing the Csound out­put for­mat will pro­duce a Csound score that will be imme­di­ate­ly con­vert­ed to an AIFF sound file dis­played on the process window:

Playing the same piece via Csound. Note that the dura­tion is 12 sec­onds (instead of 10) because a silence of 2 sec­onds (by default) is append­ed at the end of the track

Understanding the conversion process

Let us com­pare the score in com­mon Western nota­tion with its con­ver­sion to Bol Processor data. This may be help­ful for know­ing the fea­tures and lim­i­ta­tions of MusicXML files. Remember that this for­mat is a full descrip­tion of a graph­ic rep­re­sen­ta­tion of the musi­cal work. It is up to the musi­cian to add implic­it infor­ma­tion nec­es­sary for a prop­er (and artis­tic) ren­der­ing of the piece…

Musical scores of clas­si­cal works are seg­ment­ed in mea­sures marked by ver­ti­cal lines. This score con­tains 5 mea­sures of equal dura­tions. The MusicXML file con­tains data telling that the dura­tion of each mea­sure is 2 beats, mean­ing 2 sec­onds if the metronome is beat­ing at 60 beats per minute. However, instruc­tion _tempo(2) dou­bles the speed, which results in mea­sures last­ing for 1 sec­ond. The third mea­sure con­tains a chord {2, F#5, A5, D6} of half notes (min­ims) last­ing 2 beats.

The Bol Processor score also dis­plays the five mea­sures, each of which is inter­pret­ed as a poly­met­ric struc­ture. A MIDI chan­nel instruc­tion has been auto­mat­i­cal­ly insert­ed in the begin­ning of each mea­sure, indi­cat­ing which part it belongs to.

Let us read the first mea­sure and com­pare it with its con­ver­sion high­light­ed in red on the score:

{_chan(1) {2, {2, C#6}, {C#5, E5, A5} -,
{1/4, A2 C#3 E3} {1/4, A3} {3/2, A3 A3 A3}}}

The ‘2’ (green col­or) is the total dura­tion of the poly­met­ric expres­sion (or the mea­sure). On the first line is the upper score (in G key on the graph­ic score) and the sec­ond line (in F key on the image) is the low­er score. On top of the upper score is a half note C#6 inter­pret­ed as {2, C#6}. The com­ma (red col­or) indi­cates a new field of the poly­met­ric struc­ture that needs to be super­posed to the first field. It con­tains a chord {C#5, E5, A5} of quar­ter notes (crotch­ets) last­ing 1 beat.

In order to com­plete the field we need a rest of 1 beat that is not indi­cat­ed on the graph­ic score although the cor­re­spond­ing gap is men­tioned in the MusicXML file. In Bol Processor nota­tion, rests can be notat­ed ‘-’ or as inte­ger numbers/ratios. For instance, a 3-beat rest could be notat­ed “---” or {3, -}, where­as a rest of 3/4 beat should be notat­ed {3/4, -}.

The low­er score con­tains a sequence that is trou­ble­some for a machine: three grace notes “A2 C#3 E3”. Grace notes are not assigned any dura­tion in MusicXML files, so we agree with a prac­tice of giv­ing this sequence a dura­tion of exact­ly half of the prin­ci­pal note which fol­lows, here the first occur­rence of A3s declared as eight notes last­ing 1/2 beat. Consequently, the stream of grace notes has a total dura­tion of 1/4 beat and is notat­ed {1/4, A2 C#3 E3}. It is fol­lowed by A3 whose length is reduced by one half, there­fore {1/4, A3}. The fol­low­ing 3 occur­rences of A3 have a total dura­tion of 3/2 beats, hence {3/2, A3 A3 A3}.

The struc­ture of this first mea­sure is made clear on the graph­ic dis­play. Note that, unlike the pianoroll dis­play, this object rep­re­sen­ta­tion does not posi­tion sound-objects ver­ti­cal­ly accord­ing to pitch values:

The first mea­sure of the Mozart sonata’s sample

The rest of the score can be deci­phered and explained in the same man­ner. Bol Processor nota­tion is based on very sim­ple (and mul­ti­cul­tur­al) prin­ci­ples yet dif­fi­cult to cre­ate by hand… Therefore it is most con­ve­nient­ly pro­duced by gram­mars or extract­ed from MusicXML scores.

Note that it is easy to mod­i­fy the tem­po of this piece. For instance, to slow it down, insert instruc­tion _tempo(1/2) at the beginning:

Exploding scores

Clicking the EXPLODE but­ton seg­ments the musi­cal work as sep­a­rate mea­sures which make it eas­i­er to ana­lyze the con­ver­sion or reuse fragments:

The five mea­sures of Mozart’s sonata explod­ed on the Data window

Each mea­sure can be played (or expand­ed) sep­a­rate­ly. Segments are labelled [item 1], [item 2] etc. for an eas­i­er identification.

Button IMPLODE recon­structs the orig­i­nal work from its fragments.

A more complex example

Let us try DichterLiebe (op. 48) Im wun­der­schö­nen Monat Mai by Robert Schumann. The MusicXML score is in the “xml­sam­ples” fold­er dis­trib­uted in the sam­ple set “bp3-ctests-main.zip” shared on GitHub, along with its graph­ic score (read the PDF file).

The Bol Processor score is more complex:

“Im wun­der­schö­nen Monat Mai” (Robert Schumann)

This piece yields a sophis­ti­cat­ed tim­ing that can be appre­ci­at­ed on the sound output:

Im wun­der­schö­nen Monat Mai (Robert Schumann) inter­pret­ed by the Bol Processor on a PianoTeq xylophone

The cor­rect­ing ren­der­ing of this piece on Bol Processor is obtained with its (default) set­ting of quan­ti­za­tion to 10 mil­lisec­onds. Quantization is a process merg­ing the time-settings of events when these are prox­i­mate by less than a cer­tain val­ue: a human would not notice an error of 10 mil­lisec­onds in tim­ing, but merg­ing “time streaks” is an effi­cient way of sav­ing mem­o­ry space when build­ing a phase dia­gram of events. In this par­tic­u­lar piece, set­ting the quan­ti­za­tion to 30 ms already would cre­ate a notice­able default of syn­chro­niza­tion. This gives an idea of the accu­ra­cy expect­ed from human per­form­ers, which their trained audi­tive and motion­al sys­tems han­dle with­out difficulty.

Note that this MusicXML score com­pris­es 2 parts, one for voice and the sec­ond one for piano. These are sent to MIDI chan­nels 1 and 2 respec­tive­ly. These chan­nels should in turn be sent to dif­fer­ent Csound instru­ments. When sev­er­al instru­ments are not avail­able it is pos­si­ble to lis­ten to them sep­a­rate­ly by import­ing select­ed parts of the score.

Since the first mea­sure is incom­plete (1/4 beat), the piano roll is not aligned on the back­ground streaks (num­bered 0, 1, 2…):

This prob­lem can be solved by insert­ing a silence of dura­tion 3/4 in front of the score:

{3/4} {_chan(1){1/4,{{1/4,-}}},_chan(2){1/4,{{1/4,C#5},{1/4,-}}}} … etc.

which yields:

Pianoroll aligned to the time streaks

The musi­cal work may be inter­pret­ed at dif­fer­ent speeds after insert­ing a “_tempo()” instruc­tion in the begin­ning. For instance, giv­en that the metronome is set to 60 beats per minute, insert­ing _tempo(3/4) would set the tem­po to 60 * 3 / 4 = 45 beats per minute. To pro­duce a sound ren­der­ing of this par­tic­u­lar piece we insert­ed a per­for­mance con­trol _legato(25) extend­ing by 25% the dura­tions of all notes with­out mod­i­fy­ing the score. We also set up a bit of rever­ber­a­tion on the PianoTeq xylo­phone. The result­ing pianoroll was:

Same piece with _legato(25) extend­ing note dura­tions by 25%

Time-reversed Bach?

The _retro tool also gen­er­ates bizarre trans­for­ma­tions, most of which would sound “unmu­si­cal”. Some of them are inter­est­ing. For instance, this is Bach’s Goldberg Variation Nr. 5 played on Bol Processor + Csound with (Bach’s pre­sum­ably favourite) Kirnberger II tem­pera­ment — read Comparing tem­pera­ments:

Bach’s Goldberg Variation Nr. 5 (Kirnberger II tem­pera­ment) — MuseScore tran­scrip­tion by crashbangzoom808

Listen to it after apply­ing the _retro tool:

Time-reversed ver­sion of Bach’s Goldberg Variation Nr. 5 (Kirnberger II temperament)

In sum, many (musi­cal­ly mean­ing­ful) mod­i­fi­ca­tions can be achieved, includ­ing insert­ing vari­ables and send­ing the data to a gram­mar that will pro­duce entire­ly dif­fer­ent pieces. To achieve this, the gram­mar — for instance “-gr.myTransformations” — needs to be declared on top of the Data window.

The claim in favor of “well-tempered tun­ings” for inter­pret­ing Baroque music can be fur­ther assessed by com­par­ing the fol­low­ing ver­sions of J.-S. Bach’s Brandenburg Concerto Nr 2 in F major (BWV1047) part 3:

J.-S. Bach’s Brandenburg Concerto Nr 2 in F major (BWV1047) part 3 - Kirnberger II tuning
J.-S. Bach’s Brandenburg Concerto Nr 2 in F major (BWV1047) part 3 - equal-tempered tuning

Complex structures

As per this writ­ing, BP3 has been able to import and con­vert all MusicXML files con­tained in the “xml­sam­ples” fold­er. However, pieces rat­ed “too com­plex” might not be played nor expand­ed because of over­flow . Given that it is pos­si­ble to iso­late mea­sures after click­ing the EXPLODE but­ton, a PLAY safe but­ton was cre­at­ed to pick up chunks and play them in a recon­struct­ed sequence. The only draw­back is that graph­ics are deac­ti­vat­ed, which is of less­er impor­tance giv­en the com­plex­i­ty of the piece.

Listen for instance to Lee Actor’s Prelude to a Tragedy (2003), a musi­cal work made of 22 parts assigned to var­i­ous instru­ments via the 16 MIDI chan­nels — read the graph­ic score.

Lee Actor’s “Prelude to a Tragedy” (2003) with incor­rect assign­ment of some instru­ments, played by the Bol Processor using its Javascript MIDIjs play­er

The map­ping of instru­ments is faulty because most chan­nels are played as piano instead of flute, oboe, English horn, trum­pet, vio­la etc. Parts mapped to chan­nels 10 and 16 are fed with drum sounds. All these instru­ments have been syn­the­sized by the Javascript MIDIjs play­er installed on BP3’s inter­face. A bet­ter solu­tion would be to feed the “prelude-to-a-tragedy.midMIDI file to a syn­the­siz­er able to imi­tate the whole set of instru­ments, for instance MuseScore.

Score part ‘P1’: instru­ment = Picc. (V2k) — MIDI chan­nel 1
Score part ‘P2’: instru­ment = Fl. (V2k) — MIDI chan­nel 2
Score part ‘P3’: instru­ment = Ob. (V2k) — MIDI chan­nel 3
Score part ‘P4’: instru­ment = E.H. (V2k) — MIDI chan­nel 4
Score part ‘P5’: instru­ment = Clar. (V2k) — MIDI chan­nel 5
Score part ‘P6’: instru­ment = B. Cl. (V2k) — MIDI chan­nel 5
Score part ‘P7’: instru­ment = Bsn. (V2k) — MIDI chan­nel 7
Score part ‘P8’: instru­ment = Hn. (V2k) — MIDI chan­nel 8
Score part ‘P9’: instru­ment = Hn. 2 (V2k) — MIDI chan­nel 8
Score part ‘P10’: instru­ment = Tpt. (V2k) — MIDI chan­nel 9
Score part ‘P11’: instru­ment = Trb. (V2k) — MIDI chan­nel 11
Score part ‘P12’: instru­ment = B Trb. (V2k) — MIDI chan­nel 11
Score part ‘P13’: instru­ment = Tuba (V2k) — MIDI chan­nel 12
Score part ‘P14’: instru­ment = Timp. (V2k) — MIDI chan­nel 13
Score part ‘P15’: instru­ment = Splash Cymbal — MIDI chan­nel 10
Score part ‘P16’: instru­ment = Bass Drum — MIDI chan­nel 10
Score part ‘P17’: instru­ment = Harp (V2k) — MIDI chan­nel 6
Score part ‘P18’: instru­ment = Vln. (V2k) — MIDI chan­nel 14
Score part ‘P19’: instru­ment = Vln. 2 (V2k) — MIDI chan­nel 15
Score part ‘P20’: instru­ment = Va. (V2k) — MIDI chan­nel 16
Score part ‘P21’: instru­ment = Vc. (V2k) — MIDI chan­nel 16
Score part ‘P22’: instru­ment = Cb. (V2k) — MIDI chan­nel 16

Lee Actor’s “Prelude to a Tragedy” (2003) inter­pret­ed by MuseScore

Remember, though, that these meant to be raw inter­pre­ta­tions of musi­cal scores based on a few quan­ti­fied para­me­ters. To achieve a bet­ter ren­der­ing, per­for­mance para­me­ters should be insert­ed in the Bol Processor score for con­trol­ling vol­ume, panoram­ic etc. on a MIDI device, or an unlim­it­ed num­ber of para­me­ters with Csound.

Another com­plex exam­ple is Beethoven’s Fugue in B flat major (opus 133) also sup­plied as a MusicXML file. Played as a sin­gle item it takes no less than 372 sec­onds to com­pute, where­as PLAY safe deliv­ers (almost the same) in 33 seconds.

Again in this piece, the Javascript MIDIjs play­er can­not syn­the­size the two vio­lins, vio­la and cel­lo tracks (MIDI chan­nels 1 to 4). Therefore the MIDI file was sent to PianoTeq for a fair piano ren­der­ing of the mixed tracks.

Beethoven’s Fugue in B flat major — piano ver­sion played by the Bol Processor with PianoTeq

Performance controls

MusicXML files con­tain descrip­tive infor­ma­tion for use by mechan­i­cal play­back machines but not dis­played on the graph­ic score. For instance, wher­ev­er the score dis­plays “Allegretto” the file con­tains a quan­ti­ta­tive instruc­tion such as “tem­po = 132”.

Trills in mea­sure 10 of Beethoven’s Fugue in B flat major
Trills inter­pret­ed by the Bol Processor

Another notice­able case is the rep­re­sen­ta­tion of trills (see pic­ture above). These are described as sequences of fast notes in the MusicXML file. Consequently, they are accu­rate­ly ren­dered by the inter­preter of the MusicXML file.

In the same mea­sure #10, a fer­ma­ta appears on top of the crotch­et rest. Its dura­tion is not spec­i­fied because it is up to the dis­cre­tion of the per­former or con­duc­tor, but the Bol Processor fol­lows a com­mon prac­tice of mak­ing it 2 times the dura­tion of the tagged rest.

MusicXML files con­tain indi­ca­tions of sound dynam­ics which the Bol Processor inter­prets as _volume(x) com­mands. Graphic indi­ca­tions of the dynam­ics (signs ffff to pppp) are used in case a numer­ic val­ue is miss­ing. This val­ue is esti­mat­ed as per the MakeMusic Finale dynam­ics convention.

Options for import­ing a MusicXML file

Some pre­scrip­tive infor­ma­tion appear­ing on the graph­ic score is not inter­pret­ed. The first rea­son is that it would be dif­fi­cult to trans­late to Bol Processor per­for­mance con­trols — for instance stepwise/continuous vol­ume con­trol, accel­er­a­tion etc. The sec­ond rea­son is that the aim of this exer­cise is not to pro­duce the “best inter­pre­ta­tion” of a musi­cal score. Score edit­ing pro­grams per­form bet­ter! Our sole inten­tion is to cap­ture musi­cal frag­ments and rework them with gram­mars or scripts.

It would be dif­fi­cult to reuse a musi­cal frag­ment packed with strings of per­for­mance con­trols rel­e­vant to its par­tic­u­lar con­text in the musi­cal work. To this effect, the user is offered options to ignore vol­ume con­trols, tem­po and chan­nel assign­ments in every import­ed MusicXML score. These can lat­er be delet­ed or remapped in sin­gle clicks (see below).

Remapping channels and instruments

MusicXML dig­i­tal scores con­tain MIDI chan­nel spec­i­fi­ca­tions to sep­a­rate parts/instruments. These are vis­i­ble in the Bol Processor score after the con­ver­sion. In gen­er­al, they need to be remapped for the sound out­put device. MIDI chan­nels would be mod­i­fied to match instru­ments avail­able on a MIDI syn­the­siz­er, and _ins() instruc­tions may be need­ed to call instru­ments avail­able in the Csound orchestra.

This remap­ping car eas­i­ly be oper­at­ed at the bot­tom of Data or Grammar pages:

The note con­ven­tion when import­ing MusicXML scores is English (“C”, “D”, “E”…) by default. This form allows its con­ver­sion in a sin­gle click to Italian/Spanish/French (“do”, “re”, “mi”…) or Indian (“sa”, “re”, “ga”…) conventions.

Clicking the MANAGE _chan() AND _ins() but­ton dis­plays a form list­ing all occur­rences of MIDI chan­nels and Csound instru­ments. Here, for instance, we are plan­ning to keep MIDI chan­nels and insert _ins() com­mands to call Csound instru­ments described in a “-cs” Csound resource file: 

Error corrections

MuseScore’s cor­rec­tion of the defec­tive sequence (top score)

MuseScore sig­naled an error in mea­sure 142 of the MusicXML score for Beethoven’s Fugue: the total tim­ing of notes in part 1 (the upper­most score) is 3754 units which amounts to 3.66 beats (instead of 4) on a divi­sion of 1024 units per quar­ter note. MuseScore fixed this mis­take by stretch­ing this sequence to 4 beats with the mark­ing of a defec­tive silence at the end.

The Bol Processor behaves dif­fer­ent­ly. Its notion of “mea­sure” as a poly­met­ric struc­ture is not based on count­ing beats. It takes the struc­ture’s top line as ref­er­ence for tim­ing so that “mea­sures” may be of vari­able dura­tions. Its inter­pre­ta­tion of this mea­sure is the fol­low­ing: ratio 3755/1024 denotes exact­ly the (pre­sum­ably defec­tive) dura­tion of this mea­sure accord­ing to the MusicXML score:

{{3755/1024,
{{341/1024,G5}{171/512,D5}{341/512,D6 D6}{171/512,D5}{341/512,Bb5 Bb5}{171/512,A5}{341/1024,G5}{57/256,G5}{227/1024,F5}{57/256,A5}}},
{4,{{341/1024,Bb4}{171/512,Bb3}{341/512,Bb4 Bb4}{171/512,Bb3}{341/512,D5 D5}{171/512,C5}{341/512,Bb4 Bb4}{171/512,A4}{341/1024,C5}}},
{4,{{2,-}-{341/1024,-}{171/512,Eb3}{341/1024,F4}}},
{4,{{4,D4 F#3 F#3 G3 G3 E4 Eb4 Eb4}}}}

Measure # 142 inter­pret­ed by the Bol Processor

The graph­ic ren­der­ing of this mea­sure indi­cates that the four sequences are per­fect­ly synchronized.

To fix the error it is suf­fi­cient to replace “3755/1024” with “4”.

Error noti­fi­ca­tion while con­vert­ing Beethoven’s Fugue

As per this writ­ing, the Bol Processor has been able to import most MusicXML scores and play them cor­rect­ly. Errors may still occur with very com­pli­cat­ed files, notably due to incon­sis­ten­cies (or round­ing errors) in the MusicXML code. For instance, the num­ber­ing of mea­sures looks con­fus­ing in Liszt’s 14th Hungarian Rhapsody (due to implic­it mea­sures) and a few val­ues of <back­ward> are incor­rect. These errors are detect­ed and com­pen­sat­ed while con­vert­ing the file.

Liszt’s 14th Hungarian Rhapsody import­ed by the Bol Processor and played on PianoTeq — with poten­tial con­ver­sion errors
Source: ManWithNoName in the MuseScore community

Several ver­sions of the same musi­cal work may be found on reper­to­ries. Below is an inter­pre­ta­tion of the same 14th Hungarian Rhapsody based on the score edit­ed by OguzSirin:

Liszt’s 14th Hungarian Rhapsody import­ed by the Bol Processor and played on PianoTeq
Source: OguzSirin

Tempo con­trols are repro­duced in full detail when import­ing MusicXML scores. However, no more than one change per mea­sure is tak­en into account. If sev­er­al state­ments are found in a mea­sure, they may be aver­aged, or option­al­ly tak­en into account which some­times pro­duces a slight desyn­chro­niza­tion of oth­er voic­es. Discarding tem­po con­trols results in acceleration/deceleration miss­ing the del­i­ca­cy of human inter­pre­ta­tion, yet the tran­scrip­tion is clos­er to the print­ed score and its frag­ments are more eli­gi­ble for a reuse.

This sophis­ti­cat­ed exam­ple rais­es anoth­er issue. MusicXML scores con­tain tem­po state­ments of two kinds: (1) metronome pre­scrip­tive indi­ca­tions avail­able on con­ven­tion­al print­ed scores and (2) their descrip­tive mod­i­fi­ca­tions for a prop­er rend­ing of the mechan­i­cal interpretation. 

An option is offered on the Bol Processor to con­vert all tem­po state­ments or restrict to the ones of the print­ed score, which for this rhap­sody yields a pure “aca­d­e­m­ic” rendering:

Liszt’s 14th Hungarian Rhapsody inter­pret­ed by Bol Processor with exclu­sive­ly tem­po state­ments of the print­ed score
Source: ManWithNoName in the MuseScore community

Despite restric­tions (and poten­tial errors), the detailed vir­tu­os­i­ty engraved in Liszt’s score comes in sup­port to Alfred Brendel’s idea of inter­pret­ing a musi­cal work:

If I belong to a tra­di­tion, it is a tra­di­tion that makes the mas­ter­piece tell the per­former what to do, and not the per­former telling the piece what it should be like, or the com­pos­er what he ought to have composed.

Size of files

Let us com­pare the sizes of files able to deliv­er the same inter­pre­ta­tion of 14th Hungarian Rhapsody:

  • Sound file in AIFF 16-bit 48 Khz pro­duced by PianoTeq = 200 Mb
  • MusicXML file = 3.9 Mb
  • Graphic + audio score pro­duced by MuseScore = 141 Kb
  • Graphic score export­ed as PDF by MuseScore = 895 Kb
  • Csound score pro­duced by Bol Processor = 582 Kb
  • MIDI file pro­duced by Bol Processor = 75 Kb
  • Bol Processor data = 62 Kb

This com­par­i­son sup­ports the idea that Bol Processor data is arguably the most com­pact and alto­geth­er com­pre­hen­sive (text) for­mat for rep­re­sent­ing dig­i­tal music scores. Below is the full data of this musi­cal work:

// MusicXML file ‘Hungarian_Rhapsody_No._14.musicxml’ con­vert­ed
// Score part ‘P1’: instru­ment = Piano — MIDI chan­nel 1

-se.Hungarian_Rhapsody

{_tempo(14/15) _volume(82) _chan(1){9/8, 1/8 -,{1/8,C1,C2}{1/2,Db1,Db2}{1/2,E1,E2}}}{_tempo(11/10) _volume(82) _chan(1){33/8, 33/8 ‚{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/2,F1,C2,F2} 1/2 {1/8,C1,C2}{1/2,Db1,Db2}{1/2,E1,E2}}}{_tempo(11/10) _volume(82) _chan(1){33/8, 33/8 ‚{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/2,F1,C2,F2} 1/2 {1/8,C1,C2}{1/2,Db1,Db2}{1/2,E1,E2}}}{_tempo(11/10) _volume(82) _chan(1){4,{3/2,G2,B2,E3}{1/2,B2,D#3,F#3}{1/2,B2,E3,G3}{3/2,C3,E3,A3},{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/2,F1,C2,F2} 1/3 {1/6,C1,C2}{1/2,Db1,Db2}{1/2,E1,E2}}}{_tempo(11/10) _volume(82) _chan(1){4,{3/2,E3,G3,B3}{1/2,E3,A3,C4}{1/2,E3,G3,B3}{3/2,C3,E3,A3},{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/2,F1,C2,F2} 1/3 {1/6,C1,C2}{1/2,Db1,Db2}{1/2,E1,E2}}}{_tempo(11/10) _volume(82) _chan(1){4,{15/8,Bb2,Eb3,G3}{1/8,G3}{15/8,Bb2,E3,G3}{1/8,E3},{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2}{1/8,F1,C2}{1/8,F2} 1/6 {1/12,G1}{1,Ab1 B1}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3} 1/6 {1/12,C1,C2}{1/2,Db1,Db2}{1/2,E1,E2}}}{_tempo(49/60) _volume(82) _chan(1){541/120,{Ab2,C3}F3 1/8 {91/240,Ab2 C3 F3 Ab3 C4 F4 Ab4 C5 F5 Ab5 C6 F6 Ab6 C7}{1,F7 -}-1/240, 1/8 {7/8,Ab2,C3}F3{1,-Ab6}361/240, 1/8 {7/8,F1,C2}F2{1/8,F1}{91/240,C2 F2 Ab2 C3 F3 Ab3 C4 F4 Ab4 C5 F5 Ab5 C6 F6} 427/480 {53/480,G1,G2}{1/2,Ab1,Ab2}{1/2,B1,B2}, 1/8 {7/8,F1,C2}F2 5/2 }}{_tempo(11/10) _volume(82) _chan(1){4,{3/2,Eb3,G3,C4}{1/2,G3,D4}{1/2,G3,C4,Eb4}{3/2,Ab3,C4,F4},{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/2,C2,G2,C3} 1/3 {1/6,G1,G2}{1/2,Ab1,Ab2}{1/2,B1,B2}}}{_tempo(11/10) _volume(82) _chan(1){4,{3/2,C4,Eb4,G4}{1/2,Ab3,F4,Ab4}{1/2,C4,Eb4,G4}{3/2,Ab3,C4,F4},{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/2,C2,G2,C3} 1/3 {1/6,G1,G2}{1/2,Ab1,Ab2}{1/2,B1,B2}}}{_tempo(11/10) _volume(82) _chan(1){4,{15/8,G3,C4,Eb4}{1/8,D4}{15/8,F3,B3,D4}{1/8,C4},{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3}{1/8,C2,G2}{1/8,C3} 1/6 {1/12,D2}{1,Eb2 F#2}{1/8,G2,D3}{1/8,G3}{1/8,G2,D3}{1/8,G3}{1/8,G2,D3}{1/8,G3} 1/6 {1/12,G1}{1,Ab1 B1}}}{_tempo(29/30) _volume(82) _chan(1){4,{C3,Eb3,G3}C4 1/4 {3/4,C3 Eb3 G3 C4 Eb4 G4 C5 Eb5 G5 C6 Eb6 G6 C7 Eb7 G7}{1,C8 -},{C3,Eb3,G3}C4 -{1/2,Eb7}1/2,{C1,G1}{1,C2}{1/8,C1,G1}{1/8,C2}{3/4,Eb2 G2 C3 Eb3 G3 C4 Eb4 G4 C5 Eb5 G5 C6 Eb6 Ab6 C6}-,{C1,G1}C2 --}}{_tempo(29/30) _volume(82) _chan(1){4,- 3/4 {1/4,Eb4}{3/4,Eb4}{1/4,Eb4}{3/4,Eb4}{1/4,Eb4}, 4 }}{_tempo(29/30) _volume(82) _chan(1){4,{3/2,Ab4}{1,Cb5 Bb4}{3/2,Ab4},{3/2,Eb4}{1/2,Eb4}{3/4,Eb4}{1/4,Eb4}{3/4,Eb4}{1/4,Eb4},{3/2,Eb3,Cb4}{1/2,Eb3,Eb4}{1/2,Eb3,Db4}{3/2,Eb3,Cb4}}}{_tempo(29/30) _volume(82) _chan(1){4,{3/2,G4}{1,F4 Eb4}{3/2,Ab4},{1/2,Eb4}Eb4{1/2,Eb4}{3/4,Eb4}{1/4,Eb4}{3/4,Eb4}{1/4,Eb4},{3/2,Eb3,Bb3}{1/2,Eb3,Ab3}{1/2,Eb3,G3}{3/2,Eb3,Cb4}}}{_tempo(29/30) _volume(82) _chan(1){4,{7/4,Bb3,Db4,Eb4}{1/4,Ab3,Cb4,Eb4}{7/4,Ab3,Cb4,Eb4}{1/4,G3,Bb3,Eb4},{7/4,G2,Eb3}{1/4,Ab2,Eb3}{7/4,Ab2,Eb3}{1/4,Eb2,Bb2,Eb3}}}{_tempo(29/30) _volume(82) _chan(1){4,{3/2,G3,Bb3,Eb4} 1/4 {1/4,C4,C5}{3/4,C4,C5}{1/4,C4,C5}{3/4,C4,C5}{1/4,C4,C5},{3/2,Eb2,Bb2,Eb3} 1/2 --,{3/2,Eb2,Bb2,Eb3,G3,Bb3,Eb4} 5/2 }}{_tempo(29/30) _volume(82) _chan(1){4,{3/2,F4}{1,C5 G4}{3/2,F4},{3/2,C4,C5}{1/2,Ab4}{3/4,C4,C5}{1/4,C4,C5}{3/4,C4,C5}{1/4,C4,C5},{3/2,C3,Ab3}{1/2,C3,C4}{1/2,C3,Bb3}{3/2,C3,Ab3}}}{_tempo(29/30) _volume(82) _chan(1){4,{3/2,Eb4}{1,C5 C4}{3/2,F4},{3/2,C4,C5}{1/2,Db4}{3/4,C5}{1/4,C4,C5}{3/4,C4,C5}{1/4,C4,C5},{3/2,C3,G3}{1/2,C3,F3}{1/2,C3,E3}{3/2,C3,Ab3}}}{_tempo(29/30) _volume(82) _chan(1){4,{7/4,C4,C5}{1/4,C4,F4,Ab4,C5}{7/4,C4,F4,Ab4,C5}{1/4,Db4,F4,Ab4,Db5},{7/4,G4,Bb4}9/4,{7/4,E2,C3,G3,Bb3}{1/4,F2}{7/4,C3,F3,Ab3}{1/4,Ab1&}}}{_tempo(29/30) _volume(82) _chan(1){4,{15/4,Db4,F4,Ab4,Db5} 1/8 {1/8,C4,Gb4,B4,Eb5},{3/4,Ab2}{3,Db3 F3 Ab3 Db4 F4 Ab4 Db5 F5 Ab5 Db6 F6 Ab6}{1/4,-Ab1&},&Ab1 ---}}{_tempo(29/30) _volume(82) _chan(1){4,{15/4,Db4,Gb4,B4,Eb5} 1/8 {1/8,Fb4,Ab4,Cb5,Fb5},{3/4,Ab2}{3,Eb3 Gb3 B3 C4 G4 Bb4 Eb5 Gb5 B5 Eb6 Gb6 B6}{1/4,-Ab1&},&Ab1 ---}}{_tempo(29/30) _volume(82) _chan(1){4,{15/4,Fb4,Ab4,C5,Fb5} 1/8 {1/8,F4,Ab4,Db5,F5},{3/4,Ab2}{3,Fb3 Ab3 Cb4 F4 Ab4 C5 F5 Ab5 Cb6 Fb6 Ab6 Cb7}{1/4,-B1&},&Ab1 ---}}{_tempo(29/30) _volume(82) _chan(1){4,{7/2,F4,Ab4,Db5,F5} 1/2 ‚{3/4,B2}{13/4,F3 Ab3 Db4 F4 Ab4 Db5 F5 Ab5 Db6 F6 Ab6 Db7 -},&B1 ---}}{_tempo(29/30) _volume(82) _chan(1){4, 4/5 {1/5,Gb3,Db4,Gb4}{853/480,Gb3,Db4,Gb4}{107/480,F3,Ab3,Db4,F4}{F3&,Ab3&,Db4&,F4&}, 4/5 {1/5,B1,B2}{853/480,B1,B2}{107/480,C2,C3}{C2&,C3&}}}{_tempo(29/30) _volume(82) _chan(1){4,{427/480,&F3,&Ab3,&Db4,&F4}{53/480,F3,Ab3,Db4}{2,Db4}C4, 1 {F3,Ab3}{2,Eb3,G3},{427/480,&C2,&C3}{53/480,C2,Bb2,Db3}{2,Bb2,Db3}C3,-{2,C2}-}}{_tempo(9/4) _volume(113) _chan(1){4,{3/2,F4,A4,C5,F5}{1/2,G4,G5}{1/2,A4,A5}{3/2,Bb4,D5,F5,Bb5},{3/2,F2,A2,C3,F3}{1,G3 A3}{3/2,D3,F3,Bb3}}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,C5,F5,A5,C6}{1/2,D5,D6}{1/2,C5,C6}{3/2,Bb4,D5,F5,Bb5},{3/2,A2,F3,C4}{1,D4 C4}{3/2,D3,F3,Bb3}}}{_tempo(9/4) _volume(82) _chan(1){9/2, 1/8 {1/8,A4,C5,F5,A5}{2,A4,C5,F5,A5} 1/8 {1/8,G4,Bb4,E5,G5}{2,G4,Bb4,E5,G5},{1/4,C2 C3}{2,F3,A3,C4,F4}{1/4,C2 C3}{2,E3,G3,C4,E4}}}{_tempo(9/4) _volume(82) _chan(1){17/4, 1/8 {1/8,F4,A4,C5,F5}{2,F4,A4,C5,F5}--,{1/4,F1 F2}{2,C3,F3,A3,C4}--}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,C5,E5,G5,C6}{1/2,D5,D6}{1/2,E5,E6}{3/2,F5,A5,C6,F6},{1/8,C2}{11/8,C3,E3,G3,C4}{1,D4 E4}{3/2,A3,C4,F4}}}{_tempo(9/4) _volume(82) _chan(1){17/4, 1/4 {3/2,G5,C6,G6}{1/2,A5,A6}{1/2,G5,G6}{3/2,F5,A5,D6,F6},{1/4,E2 E3}{3/2,G3,C4,G4}{1,A4 G4}{3/2,F3,A3,D4,F4}}}{_tempo(9/4) _volume(82) _chan(1){9/2,{1/4,E5,G5,C6,E6}{2,E5,G5,C6,E6}{1/4,D5,F5,B5,D6}{2,D5,F5,B5,D6},{1/4,G2}{2,E3,G3,C4,E4}{1/4,G2}{2,G3,B3,D4,G4}}}{_tempo(9/4) _volume(82) _chan(1){17/4,{1/4,C5,E5,G5,C6}{2,C5,E5,G5,C6}--,{1/4,C2}{2,C3,E3,G3,C4}--}}{_tempo(9/4) _volume(113) _chan(1){4,{3/2,F4,D5,F5}{1/2,A4,F5,A5}{1/2,G4,E5,G5}{3/2,F4,D5,F5},{3/2,D3,F3,D4}{1/2,F3,A3,F4}{1/2,E3,G3,E4}{3/2,D3,F3,D4}}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,E4,C5,E5}{1/2,D4,Bb4,D5}{1/2,C4,A4,C5}{3/2,F4,A4,F5},{3/2,C3,E3,C4}{1/2,Bb2,D3,Bb3}{1/2,A2,C3,A3}{3/2,D3,F3,A3}}}{_tempo(9/4) _volume(82) _chan(1){9/2,{1/4,Bb3,G4,Bb4}{2,Bb3,G4,Bb4}{1/4,A3,F4,A4}{2,A3,F4,A4},{1/4,E2,C3,G3}{2,E2,C3,G3}{1/4,F2,D3,F3}{2,F2,C3,F3}}}{_tempo(9/4) _volume(82) _chan(1){17/4,{1/4,G3,C4,E4,G4}{2,G3,C4,E4,G4}--,{1/4,C2,G2,C3}{2,C2,G2,C3}--}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,C5,E5,C6}{1/2,D5,F5,D6}{1/2,E5,G5,E6}{3/2,F5,A5,F6},{3/2,C3,C4}{1/2,B2,B3}{1/2,Bb2,Bb3}{3/2,A2,A3}}}{_tempo(9/4) _volume(82) _chan(1){17/4, 1/4 {3/2,G5,Bb5,C6,G6}{1/2,F5,A5,F6}{1/2,E5,C6,E6}{3/2,D5,Bb5,D6},{1/4,E2 E3}{3/2,Bb3,C4,G4}{1/2,F3,A3,C4,F4}{1/2,C4,E4}{3/2,Bb2,F3,Bb3,D4}}}{_tempo(9/4) _volume(82) _chan(1){4,{C5,F5,C6}{F5,F6}{1/2,C5,C6}{3/2,Bb4,F5,Bb5},{A2,F3,C4}F4{1/2,C4}{3/2,D3,F3},{2,C4} 1/2 {3/2,Bb3}}}{_tempo(9/4) _volume(82) _chan(1){9/2, 1/8 {1/8,A4,C5,F5,A5}{2,A4,C5,F5,A5} 1/8 {1/8,G4,Bb4,E5,G5}{2,G4,Bb4,E5,G5},{1/4,C2 C3}{2,F3,A3,C4,F4}{1/4,C2 C3}{2,E3,G3,C4,E4}}}{_tempo(9/4) _volume(82) _chan(1){17/4, 1/8 {1/8,F4,A4,C5,F5}{2,F4,A4,C5,F5}--,{1/4,F1 F2}{2,C3,F3,A3,C4}--}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,F4,D5,F5}{1/2,A4,F5,A5}{1/2,G4,E5,G5}{3/2,F4,D5,F5},{3/2,D3,F3,D4}{1/2,F3,A3,F4}{1/2,E3,G3,E4}{3/2,D3,F3,D4}}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,E4,C5,E5}{1/2,D4,Bb4,D5}{1/2,C4,A4,C5}{3/2,F4,A4,F5},{3/2,C3,E3,C4}{1/2,Bb2,D3,Bb3}{1/2,A2,C3,A3}{3/2,D3,F3,A3}}}{_tempo(9/4) _volume(82) _chan(1){9/2,{1/4,Bb3,G4,Bb4}{2,Bb3,G4,Bb4}{1/4,A3,D4,F#4,A4}{2,A3,D4,F#4,A4},{1/4,G2,C3,G3}{2,G2,C3,G3}{1/4,D2,F#2,D3}{2,D2,F#2,D3}}}{_tempo(9/4) _volume(82) _chan(1){17/4,{1/4,G3,Eb4,G4}{3,G3,Eb4,G4}-,{1/4,Eb2,Bb2,Eb3}{3,Eb2,Bb2,Eb3}-}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,C5,Eb5,Ab5,C6}{1/2,D5,F5,Bb5,D6}{1/2,Eb5,G5,C6,Eb6}{3/2,F5,Ab5,Db6,F6},{3/2,A2,Eb3,Ab3,C4}{1/2,F3,B3,D4}{1/2,Ab3,C4,Eb4}{3/2,Ab3,Db4,F4}}}{_tempo(9/4) _volume(82) _chan(1){4,{3/2,G5,Bb5,C6,G6}{1/2,F5,A5,F6}{1/2,E5,C6,E6}{3/2,D5,Bb5,D6},{3/2,G3,Bb3,C4,G4}{1/2,F3,A3,C4,F4}{1/2,C4,E4}{3/2,Bb2,F3,Bb3,D4}}}{_tempo(9/4) _volume(82) _chan(1){4,{C5,F5,C6}{F5,F6}{1/2,C5,C6}{3/2,Bb4,F5,Bb5},{A2,F3,C4}F4{1/2,C4}{3/2,D3,F3},{2,C4} 1/2 {3/2,Bb3}}}{_tempo(9/4) _volume(82) _chan(1){9/2, 1/8 {1/8,A4,C5,F5,A5}{2,A4,C5,F5,A5} 1/8 {1/8,G4,Bb4,E5,G5}{2,G4,Bb4,E5,G5},{1/4,C2 C3}{2,F3,A3,C4,F4}{1/4,C2 C3}{2,E3,G3,C4,E4}}}{_tempo(9/4) _volume(82) _chan(1){17/4, 1/8 {1/8,F4,A4,C5,F5}{2,F4,A4,C5,F5}--,{1/4,F1 F2}{2,C3,F3,A3,C4}--}}{_tempo(9/4) _volume(113) _chan(1){9/2, 1 {1/2,A3,C4,F4}{1/2,A4,C5,F5}{1/2,G5,C6,E6,G6}{1/2,A5,C6,F6,A6}{3/2,Bb5,D6,F6,Bb6},{1/2,C3 D3 E3}{1/2,F3}{1/2,A2,C3,F3}{1/2,A3,C4,F4}{1/2,G3,C4,E4,G4}{1/2,A3,C4,F4,A4}{3/2,Bb3,D4,F4,Bb4},{1/2,C2 D2 E2}{1/2,F1,F2} 7/2 }}{_tempo(9/4) _volume(82) _chan(1){9/2, 1 {1/2,C4,F4,A4,C5}{1/2,C5,F5,A5,C6}{1/2,D6,F6,Bb6,D7}{1/2,C6,F6,A6,C7}{3/2,Bb5,D6,F6,Bb6},{1/2,C3 D3 E3}{1/2,F3}{1/2,C3,F3,A3}{1/2,C4,F4,A4}{1/2,D4,F4,Bb4,D5}{1/2,C4,F4,A4,C5}{3/2,Bb3,D4,F4,Bb4},{1/2,C2 D2 E2}{1/2,F1,F2} 7/2 }}{_tempo(9/4) _volume(82) _chan(1){9/2, 1 {1/2,A5,C6,F6,A6}{A5,C6,F6,A6} 1/2 {1/2,G5,Bb5,E6,G6}{G5,Bb5,E6,G6},{1/2,C3 D3 E3}{1/2,F3}{1/2,C3,F3,A3,C4}{4/5,A3,C4,F4,A4}{1/5,B1,B2}{1/2,C2,C3}{1/2,C3,E3,G3,C4}{G3,C4,E4,G4},{1/2,C2 D2 E2}{1/2,F1,F2} 7/2 }}{_tempo(9/4) _volume(82) _chan(1){33/8, 1/8 1/2 {1/2,A3,C4,F4}{1/2,A4,C5,F5}{1/2,A5,C6,F6}{1/2,F6,A6,C7,F7} 1/2 -,{1/8,E1,E2}{1/2,F1,F2}{1/2,A2,C3,F3}{1/2,A3,C4,F4}{1/2,A3,C4,F4}{1/2,F4,A4,C5,F5} 1/2 -}}{_tempo(9/4) _volume(82) _chan(1){9/2, 1 {1/2,E4,G4,C5}{1/2,E5,G5,C6}{1/2,D6,G6,B6,D7}{1/2,E6,G6,C7,E7}{3/2,F6,A6,C7,F7},{1/2,G3 A3 B3}{1/2,C4}{1/2,E3,G3,C4}{1/2,E4,G4,C5}{1/2,G3,B3,D4,G4}{1/2,C4,E4,G4,C5}{3/2,A3,C4,F4,A4},{1/2,G2 A2 B2}{1/2,C2,C3} 7/2 }}{_tempo(9/4) _volume(82) _chan(1){9/2, 1 {1/2,C4,F4,A4,C5}{1/2,C5,F5,A5,C6}{1/2,D6,F6,Bb6,D7}{1/2,C6,F6,A6,C7}{3/2,Bb5,D6,F6,Bb6},{1/2,G3 A3 B3}{1/2,C4}{1/2,G3,C4,E4}{1/2,G4,C5,E5}{1/2,F4,A4,D5,F5}{1/2,C4,E4,G4,C5}{3/2,F4,A4,C5,F5},{1/2,G2 A2 B2}{1/2,C2,C3} 7/2 }}{_tempo(9/4) _volume(82) _chan(1){9/2, 1 {1/2,E5,G5,C6,E6}{E6,G6,C7,E7} 1/2 {1/2,D5,F5,B5,D6}{D6,F6,B6,D7},{1/2,G3 A3 B3}{1/2,C4}{1/2,G3,C4,E4,G4}{4/5,E4,G4,C5,E5}{1/5,F#2,F#3}{1/2,G2,G3}{1/2,G3,B3,D4,G4}{D4,G4,B4,D5},{1/2,G2 A2 B2}{1/2,C2,C3} 7/2 }}{_tempo(9/4) _volume(124) _chan(1){33/8, 1/8 1/2 {1/2,E3,G3,C4}{1/2,E4,G4,C5}{1/2,E5,G5,C6}{1/2,C6,E6,G6,C7} 1/2 -,{1/8,B1,B2}{1/2,C2,C3}{1/2,E2,G2,C3}{1/2,E3,G3,C4}{1/2,E4,G4,C5}{1/2,C4,E4,G4,C5} 1/2 -}}{_tempo(2) _volume(97) _chan(1){4, 1/2 {1/6,C5}{1/6,F5,C6}{1/6,C6}{1/6,F6,C7}{1/6,C6}{1/6,F5,C6}{1/6,C5}{1/6,A5,C6}{1/6,C6}{1/6,G6,C7}{1/6,C6}{1/6,G5,C6}{1/6,C5}{1/6,F5,C6}{1/6,C6}{1/2,F6,C7} 1/2 ‚{3/2,A3,C4,F4}{1/2,C4,A4}{1/2,Bb3,C4,G4}{3/2,A3,C4,F4}}}{_tempo(2) _volume(82) _chan(1){4, 1/2 {1/6,C5}{1/6,E5,C6}{1/6,C6}{1/6,E6,C7}{1/6,C6}{1/6,E5,C6}{1/2,C5 C6 C6}{1/6,G6,C7}{1/2,C6 C6 C5}{1/6,F5,C6}{1/6,C6}{1/2,F6,C7} 1/2 ‚{3/2,G3,C4,E4}{1/2,F3,C4,D4}{1/2,E3,C4}{3/2,A3,C4,F4}}}{_tempo(2) _volume(82) _chan(1){4,{1/2,C4,G4,Bb4} 1/2 {1/6,G6,Bb6}{1/3,E6 C6}{1/6,G5,Bb5}{1/3,E5 C5}{1/2,F4,A4} 1/2 {1/6,F6,A6}{1/3,D6 Bb5}{1/6,F5,A5}{1/3,D5 Bb4},{1,E2 C3 E3 G3 Bb3 C4}{1/2,G4,Bb4} 1/2 {1,F2 C3 F3 A3 C4 F4}{1,A4 -}}}{_tempo(2) _volume(82) _chan(1){4,{1/2,E4,G4} 1/2 {1/6,E6,G6}{1/3,C6 G5}{1/6,G5,C6}{1/3,E5 C5}{1/6,E5,G5}{1/3,C5 G4}{1/6,G4,C5}{1/3,E4 C4}{1/6,E4,G4}{1/3,C4 G3} 1/2 ‚{1,C2 G2 C3 E3 G3 C4}{3/2,G4 --}-{1/2,E3 C3 G2}}}{_tempo(2) _volume(82) _chan(1){4, 1/2 {1/2,E4,G4,C5}{1/2,E5,G5,C6}{1/2,D6,F6,D7}{1/2,E6,G6,E7}{3/2,F6,A6,F7},{1/2,C1}{1/2,E3,G3,C4}{1/2,E4,G4,C5}{1/2,D4,F4,B4,D5}{1/2,E4,G4,Bb4,E5}{3/2,F4,A4,F5}}}{_tempo(2) _volume(82) _chan(1){4,{1/2,G3,Bb3,G4}{1/2,G4,Bb4,G5}{1/2,G6,Bb6,G7}{1/2,F6,A6,F7}{1/2,E6,C7,E7}{3/2,D5,Bb5,D6},{1/2,E2,A2,C3}{1/2,E3,C4,E4}{1/2,E4,C5,E5}{1/2,F4,A4,F5}{1/2,C5,E5}{3/2,Bb3,F4,Bb4,D5}}}{_tempo(2) _volume(82) _chan(1){4,{1/3,C6,F6,C7}{1/3,F6,F7}{1/3,C6,C7}{1/3,F6,F7}{1/3,C6,C7}{1/3,F5,F6}{1/3,C5,C6}{1/3,F4,F5}{1/3,C4,C5}{Bb3,F4,Bb4},{3,A5 D6 C5 F5 C5 F4 C4 F3 C3}{D2,Bb2},{F4,D5}---}}{_tempo(2) _volume(82) _chan(1){4,{1/2,A3,F4,A4}{1/2,A4,F5,A5}{4/5,A5,F6,A6}{1/5,G3,C4,G4}{1/2,G3,C4,G4}{1/2,G4,C5,G5}{G5,C6,G6},{1/2,C2,A2,C3}{1/2,C3,A3,C4}{4/5,C4,F4,A4,C5}{1/5,C2,E2,G2,C3}{1/2,C2,E2,G2,C3}{1/2,C3,E3,G3,C4}{C4,E4,G4,C5}}}{_tempo(2) _volume(82) _chan(1){33/8,{1/8,F3,A3,C4,F4}{1/2,F3,A3,C4,F4}{1/2,F4,A4,C5,F5}{1/2,F5,A5,C6,F6}{1/2,F6&,A6&,C7&,F7&}{&F6,&A6,&C7,&F7}-,{1/8,F1,A1,C2,F2}{1/2,F1,A1,C2,F2}{1/2,F2,A2,C3,F3}{1/2,F3,A3,C4,F4}{1/2,F4&,A4&,C5&,F5&}{&F4,&A4,&C5,&F5}-}}{_tempo(5/3) _volume(97) _chan(1){4, 1/2 {7/2,C5 C6 D6 C6 C7 D7 C7 C6 D6 C6 C5 C6 D6 C6 C7 D7 C7 C6 D6 C6 C5 C6 D6 C6 C7 D7 C7 C6},{3/2,A3,C4,F4}{1/2,C4,A4}{1/2,Bb3,C4,G4}{3/2,A3,C4,F4}}}{_tempo(5/3) _volume(82) _chan(1){4,{4,D6 C6 C5 C6 D6 C6 C7 D7 C7 C6 D6 C6 C5 C6 D6 C6 C7 D7 C7 C6 D6 C6 C5 C6 D6 C6 C7 D7 C7 C6 F6 A6},{3/2,G3,C4,E4}{1/2,F3,C4,D4}{1/2,E3,C4}{3/2,A3,C4,F4}}}{_tempo(4/3) _volume(82) _chan(1){643/160,{2,D7 C#7 C7 B6 Bb6 A6 Ab6 G6 F#6 F6 E6 Eb6 D6 C#6 C6 B5 Bb5 A5 Ab5 G5}{323/160,F#5 F5 E5 Eb5 D5 C#5 C5 B4 Bb4 A4 Ab4 G4 F#4 F4 E4 Eb4 D4 C#4 D4}, 1/8 {15/8,Bb2,G3,D4}{2,D3,A3,C4,F#4}}}{_tempo(4/3) _volume(82) _chan(1){33/8,{1/8,G3,E4,G4}{4,D4 C#4 C4 B3 Bb3 A3 Ab3 G3 F#3 F3 E3 Eb3 D3 Db3 C3 B2 Bb2 A2 Ab2 G2 F#2 F2 E2 Eb2 E2 F2 F#2 G2 Ab2 A2 Bb2 B2},{1/8,G2,E3,G3}{4,D3 C#3 C3 B2 Bb2 A2 Ab2 G2 F#2 F2 E2 Eb2 D2 Db2 C2 B1 Bb1 A1 Ab1 G1 F#1 F1 E1 Eb1 E1 F1 F#1 G1 Ab1 A1 Bb1 B1}}}{_tempo(5/3) _volume(113) _chan(1){4,{1/2,C3}{1/2,C4,E4,G4,C5}{1/2,C5,E5,G5,C6}{1/2,D6,F6,D7}{1/2,E6,G6,E7}{3/2,F6,A6,F7},{1/2,C1,C2}{1/2,C3,E3,G3,C4}{1/2,C4,E4,G4,C5}{1/2,B3,Ab4,B4}{1/2,Bb3,G4,Bb4}{3/2,A3,F4,A4}}}{_tempo(5/3) _volume(82) _chan(1){4,{1/2,G3,Bb3,G4}{1/2,G4,Bb4,G5}{1/2,G5,Bb5,G6}{1/2,F6,A6,F7}{1/2,E6,C7,E7}{3/2,D6,Bb6,D7},{1/2,E2,A2,C3}{1/2,E3,C4,E4}{1/2,E4,C5,E5}{1/2,F5,A5,C6,F6}{1/2,C6,E6}{3/2,Bb4,F5,Bb5,D6}}}{_tempo(5/3) _volume(82) _chan(1){4,{1/3,C6,F6,C7}{1/3,F6,F7}{1/3,C6,C7}{1/3,F6,F7}{1/3,C6,C7}{1/3,F5,F6}{1/3,C5,C6}{1/3,F4,F5}{1/3,C4,C5}{Bb3,F4,Bb4},{1/3,F4,D5,A5}{8/3,D6 C5 F5 C5 F4 C4 F3 C3}{D2,Bb2}}}{_tempo(5/3) _volume(124) _chan(1){4,{1/2,A3,F4,A4}{1/2,A4,F5,A5}{7/8,A5,F6,A6}{1/8,G3,C4,G4}{1/2,G3,C4,G4}{1/2,G4,C5,G5}{7/8,G5,C6,G6}{1/8,F3,A3,C4,F4},{1/2,C2,A2,C3}{1/2,C3,A3,C4}{7/8,C4,A4,C5}{1/8,C2,E2,G2,C3}{1/2,C2,E2,G2,C3}{1/2,C3,E3,G3,C4}{7/8,C4,E4,G4,C5}{1/8,F1,A1,C2,F2}}}{_tempo(5/3) _volume(82) _chan(1){4,{1/2,F3,A3,C4,F4}{1/2,F4,A4,C5,F5}{1/2,F5,A5,C6,F6}{1/2,F6&,A6&,C7&,F7&}{&F6,&A6,&C7,&F7}-,{1/2,F1,A1,C2,F2}{1/2,F2,A2,C3,F3}{1/2,F3,A3,C4,F4}{1/2,F4&,A4&,C5&,F5&}{&F4,&A4,&C5,&F5}-, 1/64 7/64 }}{_tempo(7/4) _volume(52) _chan(1){1/2,{1/2,A3&}, 1/2 }}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,&A3,D4,A4}{1/2,A3,D4,A4}{1/2,A3,D4,A4}{1/4,A3,D4,G#4}{1/4,B4},{1/2,D2,F#3}{1/2,A2,F#3}{1/2,D3,F#3}{1/2,A2,F#3}}}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,A4}{1/2,G4 B4}{1,A4 D4&},{1/2,A3,D4}{1/2,A3,D4}{1/2,A3,D4}1/2,{1/2,D2,F#3}{1/2,A2,F#3}{1/2,D3,F#3} 1/2 }}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,&D4,F#4,C5}{1/2,D4,F#4,C5}{1/2,D4,F#4,C5}{1/2,B4 D5}, 3/2 {1/2,D4,F#4},{1/2,D2,A3}{1/2,A2,A3}{1/2,D3,A3}{1/2,A2,A3}}}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,C5}{1/2,B4 D5}{1,C5 D4&},{1/2,D4,F#4}{1/2,D4,F#4}{1/2,D4,F#4}1/2,{1/2,D2,A3}{1/2,A2,A3}{1/2,D3,A3} 1/2 }}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,&D4,G4,B4}{1/2,D4,G4,B4}{1/2,D4,G4,B4}{1/2,A#4 C5}, 3/2 {1/2,D4,F#4},{1/2,G2,B3}{1/2,D3,B3}{1/2,G3,B3}{1/2,D3,B3}}}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,B4}{1/2,A#4 C5}B4,{1/2,D4,G4}{1/2,D4,F#4}{D4,G4},{1/2,G2,B3}{1/2,D3,B3}{G3,B3}}}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,B4,G5,B5}{1/2,B4,G5,B5}{1/2,B4,G5,B5}{1/2,A#4,F#5}, 3/2 {1/2,A#5 C6},{1/2,G2,D4}{1/2,D3,D4}{1/2,B3,D4}{1/2,D3,D4}}}{_tempo(5/3) _volume(82) _chan(1){2,{1/2,B4,G5}{1/2,A#4,F#5}{B4,G5},{1/2,B5}{1/2,A#5 C6}B5,{1/2,G2,D4}{1/2,D3,D4}{G3,D4}}}{_tempo(2) _volume(52) _chan(1){4,{3/2,D5,B5}{1/2,D5,D6}{1/2,D5,C#6}{3/2,D5,B5},{3/2,G4}{1,B4 A4}{3/2,G4}}}{_tempo(7/15) _volume(82) _chan(1){961/240,{3/2,D5,A5}{1/2,D5,G5}{1/2,D5,F#5}{3/2,G5 --}1/240, 5/2 {1/2,C#5}{77/80,F#5 G5 A#5 B5 E6 E#6 F#6 A6 G6 E6 C#6 A#5 G5 E5 E#5 F#5 A5 G5 B#4 C#5 F#5 E5}{1/24,-D5}, 1/8 {11/8,F#4,A4}{1/2,E4,A4}{1/2,D4,A4}{3/2,A#3,E4,G4}}}{_tempo(7/3) _volume(97) _chan(1){2,B4{1,-D4},D5 1,{B3,F#4} 1 }}{_tempo(7/3) _volume(82) _chan(1){2,{1,F#4 G4 A4 B4}{1,A4 C#5},{1,D4}{1/2,D4,F#4}{1/2,C#4,G4},{A2,F#3,A3}{1/2,A2,F#3,A3}{1/2,A2,E3,A3}}}{_tempo(7/3) _volume(82) _chan(1){2,D5{1,-D5},{D4,F#4}1,{D3,A3}-}}{_tempo(7/3) _volume(82) _chan(1){2,{1,F#5 G5 A5 B5}{1,A5 C#6},{1,D5}{1/2,D5,F#5}{1/2,C#5,G5},{A3,F#4,A4}{1/2,A3,F#4,A4}{1/2,A3,E4,A4}}}{_tempo(7/3) _volume(82) _chan(1){2,D6{1,-A5},{D5,F#5}1,{D4,A4} 1 }}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,A5,F#6,A6}{1/2,A5,F#6,A6}{1/2,A5,F#6,A6}{1/2,G#6 B6}, 3/2 {1/2,G#5,E#6},{1/2,G2,G3}{1/2,A3,F#4,A4}{1/2,C#4,F#4,A4}{1/2,A3,F#4,A4}}}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,A6}{1/2,G#6 B6}{1,A6 D6},{1/2,A5,F#6}{1/2,G#5,E#6}{1/2,A5,F#6}1/2,{1/2,D4,F#4,A4}{1/2,A3,F#4,A4}{1/2,D4,F#4,A4} 1/2 }}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,C6,A6,C7}{1/2,C6,A6,C7}{1/2,C6,A6,C7}{1/2,B6 D7}, 3/2 {1/2,B5,G#6},{1/2,D2,D3}{1/2,C4,A4,C5}{1/2,E4,A4,C5}{1/2,C4,A4,C5}}}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,C7}{1/2,B6 D7}{1,C7 F#6},{1/2,C6,A6}{1/2,B5,G#6}{1/2,C6,A6}1/2,{1/2,F#4,A4,C5}{1/2,C4,A4,C5}{1/2,F#4,A4,C5} 1/2 }}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,B5,G6,B6}{1/2,B5,G6,B6}{1/2,B5,G6,B6}{1/2,A#6 C7}, 3/2 {1/2,A#5,F#6},{1/2,G2,G3}{1/2,B3,G4,B4}{1/2,D4,G4,B4}{1/2,B3,G4,B4}}}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,B6}{1/2,A#6 C7}B6,{1/2,B5,G6}{1/2,A#5,F#6}{B5,G6},{1/2,D4,G4,B4}{1/2,B3,G4,B4}{D4,G4,B4}}}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,D6,B6,D7}{1/2,D6,B6,D7}{1/2,D6,B6,D7}{1/2,C#7 E7}, 3/2 {1/2,C#6,A#6},{1/2,G2,G3}{1/2,D4,B4,D5}{1/2,F#4,B4,D5}{1/2,D4,B4,D5}}}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,D7}{1/2,C#7 E7}{1,D7 -},{1/2,D6,B6}{1/2,C#6,A#6}{1/2,D6,B6}1/2,{1/2,G4,B4,D5}{1/2,D4,B4,D5}{1/2,G4,B4,D5} 1/2 }}{_tempo(7/4) _volume(113) _chan(1){2,{1/2,C#4,D4,F#4,A4}{1/2,C#4,D4,F#4,A4}{1/2,C#4,D4,F#4,A4}{1/2,A#3 C4},{1/2,B1,B2}{1/2,B1,B2}{1/2,B1,B2}{1/2,A#2 C3}}}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,C4,D4,F#4,A4}{1/2,A#3 C4}{1/2,C4,D4,F#4,A4} 1/2 ‚{1/2,B1,B2}{1/2,A#2 C3}{1/2,B1,B2} 1/2 }}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,C#4,D4,F#4,A4,D5}{1/2,C#4,D4,F#4,A4,D5}{1/2,C#4,D4,F#4,A4,D5}{1/2,A#3 C4},{1/2,B1,B2}{1/2,B1,B2}{1/2,B1,B2}{1/2,A#2 C3}}}{_tempo(7/4) _volume(82) _chan(1){2,{1/2,A3,B3,D#4,F#4,B4}{1/2,A#3 C4}{1/2,A3,B3,D#4,F#4,B4} 1/2 ‚{1/2,G1,G2}{1/2,A#2 C3}{1/2,B1,B2} 1/2 }}{_tempo(7/3) _volume(97) _chan(1){2,{1/2,B4,G#5,B5}{1/2,B4,G#5,B5}{1/2,B4,G#5,B5}{1/2,A#4 F5 C#6},{1/2,E2,E3}{1/2,G#3,E4}{1/2,B3,G#4}{1/2,G#3,E4}}}{_tempo(7/3) _volume(82) _chan(1){2,{1/2,B4,G#5,B5}{1/2,A#4 F5 C#6}{1/2,B4,G#5,B5} 1/2 ‚{1/2,E2,E3}{1/2,G#3,E4}{1/2,B3,G#4}{1/2,G#3,E4}}}{_tempo(7/3) _volume(97) _chan(1){2,{1/2,D5,B5,D6}{1/2,D5,B5,D6}{1/2,D5,B5,D6}{1/2,C#5 A#5 E6},{1/2,E2,E3}{1/2,B3,G#4}{1/2,D4,B4}{1/2,B3,G#4}}}{_tempo(7/3) _volume(82) _chan(1){2,{1/2,D5,B5,D6}{1/2,C#5 A#5 E6}{1/2,D5,B5,D6} 1/2 ‚{1/2,E2,E3}{1/2,B3,G#4}{1/2,D4,B4}{1/2,B3,G#4}}}{_tempo(7/3) _volume(82) _chan(1){2,{1/2,C#5,A5,C#6}{1/2,C#5,A5,C#6}{1/2,C#5,A5,C#6}{1/2,B#4 G#5 D6},{1/2,A2,A3}{1/2,A3,E4}{1/2,C#4,A4}{1/2,A3,E4}}}{_tempo(7/3) _volume(82) _chan(1){2,{1/2,C#5,A5,C#6}{1/2,B#4 G#5 D6}{1/2,C#5,A5,C#6} 1/2 ‚{1/2,A2,A3}{1/2,A3,E4}{1/2,C#4,A4}{1/2,A3,E4}}}{_tempo(7/3) _volume(82) _chan(1){2,{1/2,E5,C#6,E6}{1/2,E5,C#6,E6}{1/2,E5,C#6,E6}{1/2,D#5 B5 F#6},{1/2,A2,A3}{1/2,C#4,A4}{1/2,E4,C#5}{1/2,C#4,A4}}}{_tempo(7/3) _volume(82) _chan(1){2,{1/2,E5,C#6,E6}{1/2,D#5 B5 F#6}{1/2,E5,C#6,E6} 1/2 ‚{1/2,A2,A3}{1/2,C#4,A4}{1/2,E4,C#5} 1/2 }}{_tempo(7/3) _volume(52) _chan(1){4,{3/2,E5,C#6}{1/2,E5,E6}{1/2,E5,D#6}{3/2,E5,C#6},{3/2,G4}{1,B4 A4}{3/2,G4}}}{_tempo(7/15) _volume(82) _chan(1){321/80,{3/2,D5,A5}{1/2,D5,G5}{1/2,D5,F#5}{1/8,B5}{3/8,A5}{39/40,G5 A5 B#5 D6 F#6 F6 G6 B6 A6 F#6 D6 B#5 A5 F#5 D5 E5 E#5 F#5 F5 G5 B5 A5 C5 D5 G5 F#5}{3/80,-E5}, 1/8 {11/8,F#4,A4}{1/2,E4,A4}{1/2,D4,A4}{1/2,A#3,E4,G4}-}}{_tempo(127/60) _volume(97) _chan(1){2,{C#5,E5}{1,-E4},{C#4,G#4} 1 }}{_tempo(127/60) _volume(82) _chan(1){2,{1,G#4 A4 B4 C#5}{1,B4 D#5},{1,E4}{1/2,E4,G#4}{1/2,D#4,A4},{B2,G#3,B3}{1/2,B2,G#3,B3}{1/2,B2,F#3,B3}}}{_tempo(127/60) _volume(82) _chan(1){2,{E4,G#4,E5}{1,-E5},{E3,B3}-}}{_tempo(127/60) _volume(82) _chan(1){2,{1,G#5 A5 B5 C#6}{1/2,E5,B5}{1/2,D#5,A5,D#6},E5 1,{B3,G#4}{1/2,B3,G#4,B4}{1/2,B3,F#4,B4}}}{_tempo(127/60) _volume(82) _chan(1){2,{E5,G#5,E6}-,{E4,B4}{1,-E2}}}{_tempo(127/60) _volume(82) _chan(1){2,{B3,E4,B4}{1/2,B3,E4,G#4}{1/2,A3,D#4},{1,G#2 A2 B2 C#3}{1,B2 B1}}}{_tempo(127/60) _volume(82) _chan(1){2,{G#3,E4}-,E2 -}}{_tempo(7/4) _volume(82) _chan(1){2,{1/8,D#5}{3/8,E5&}{1/4,&E5,D6} 1/4 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1/2 , 1/3 {1/6,A6 B6}3/2,{3/8,F5}{1/8,C4}{3/8,A4}{1/8,E4}{1/2,A3,C4,E4,A4} 1/2 }}{_tempo(12/5) _volume(124) _chan(1){9/4,{1/4,A4,F5,A5}{A4,F5,A5}{Bb4,E5,Bb5},{1/4,F3,C4,F4}{F3,C4,F4}{C3,G3,C4}}}{_tempo(12/5) _volume(82) _chan(1){2,{3/2,A4,F5,A5}{1/4,A5,F6}{1/4,Bb5,D6},{3/2,F2,C3,F3} 1/2 }}{_tempo(12/5) _volume(82) _chan(1){2,{1/4,A5,F6}{1/4,Bb5,D6}{1/4,Bb5,D6}{1/4,A5,C6}{1/4,A5,C6}{1/4,G5,Bb5}{1/4,G5,Bb5}{1/4,G#5,B5},{1/2,C2,C3}{1/2,G3,Bb3,E4}{1/2,G3,Bb3,E4}{1/2,G3,Bb3,E4}}}{_tempo(12/5) _volume(82) _chan(1){2,{1/4,Bb5,D6}{1/4,A5,C6}{1/4,A5,C6}{1/4,G5,Bb5}{F5,A5},{1/2,F2,F3}{1/2,A3,C4,F4}{A3,C4,F4}}}{_tempo(12/5) _volume(124) _chan(1){9/4,{1/4,A4,F5,A5}{A4,F5,A5}{Bb4,E5,Bb5},{1/4,F3,C4,F4}{F3,C4,F4}{C3,G3,C4}}}{_tempo(12/5) _volume(82) _chan(1){2,{A4,F5,A5}{1,-A5 A6 G#5},{3/2,F2,C3,F3} 1/2 }}{_tempo(12/5) _volume(82) _chan(1){2,{2,G#6 G5 G6 F5 F6 E5 E6 D5},{1/2,A2}{1/2,D4,F4}{1/2,A3}{1/2,C#4,G4}}}{_tempo(12/5) _volume(82) _chan(1){2,{1,D6 C#5 C#6 D5&}{1/2,&D5,D6} 1/2 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{1/8,E5&}{3/8,&E5,D6}{1/8,E5&}{3/8,&E5,D6},{3/8,G#3}{1/8,D4}{3/8,B4}{1/8,E4}{3/8,E3}{1/8,B3}{3/8,G#4}{1/8,E4}}}{_tempo(8/5) _volume(82) _chan(1){2,{1/8,E5}{1/4,D6}{1/8,C6}{3/8,B5}{1/8,A5}{3/8,G#5}{1/8,E5}{3/8,F#5}{1/8,G#5},{3/8,A3}{1/8,D4}{3/8,B4}{1/8,E4}{3/8,E3}{1/8,B3}{3/8,G#4}{1/8,E4}}}{_tempo(8/5) _volume(82) _chan(1){2,{1/8,B5}{1/4,A5}{1/8,G#5}{3/8,A5}{1/8,B5}{1/8,D#6}{1/4,C6}{1/8,E5}{3/8,F#5}{1/8,G#5},{3/8,G#3}{1/8,D4}{3/8,B4}{1/8,E4}{3/8,E3}{1/8,D4}{3/8,B4}{1/8,E4}}}{_tempo(8/5) _volume(82) _chan(1){2,{1/3,A5}{1/6,A5 B5}{3/8,A5}{1/8,G#5}A5,{3/8,A3}{1/8,C4}{3/8,A4}{1/8,E4}{3/8,A3,C4}{1/8,D4}{1/2,A4}}}{_tempo(8/5) _volume(52) _chan(1){2,{2,D#6 E6 F6 E6 D7 E6 F6 E6 D7 E6 F6 E6 D7 E6 F6 E6},{2,-D7 D7 D7},{3/8,G#3}{1/8,D4}{3/8,B4}{1/8,E4}{3/8,E3}{1/8,B3}{3/8,G#4}{1/8,E4}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,D7 E6 C7 E6 B6 E6 A6 E6 G#6 E6 A6 E6 B6 E6 G#6 E6},{2,D7 C7 B6 A6 G#6 A6 B6 G#6},{3/8,G#3}{1/8,D4}{3/8,B4}{1/8,E4}{3/8,E3}{1/8,B3}{3/8,G#4}{1/8,E4}}}{_tempo(8/5) _volume(82) 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_chan(1){2,{2,A6 E6 C#7 E6 E7 E6 C#7 E6 A6 E6 C#7 E6 E7 E6 C#7 E6},{A#3,C4,A4} 427/480 {53/480,A4}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 E6 C#7 E6 E7 E6 C#7 E6 A6 E6 C#7 E6 E7 E6 C#7 E6},{2,A4 G5 G5 G5}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 E6 C#7 E6 E7 E6 C#7 E6 A6 E6 C#7 E6 E7 E6 C#7 E6},{3/8,G5}{1/8,F5}{3/8,E5}{1/8,D5}{3/8,C#5}{1/8,D5}{3/8,E5}{1/8,C5}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 F6 D7 F6 F7 F6 D7 F6 A6 F6 D7 F6 F7 F6 D7 F6},{1/8,E5}{1/4,D5}{1/8,C#5}{3/8,D5}{1/8,E5}{1/8,G5}{1/4,F5}{1/8,E5}{3/8,F5}{1/8,G5}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 F6 D7 F6 F7 F6 D7 F6 A6 F6 D7 F6 F7 F6 D7 F6},{3/8,G#5}{1/8,A5}{3/8,Bb5}{1/8,A5}{3/8,G5}{1/8,F5}{3/8,E5}{1/8,D5}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,G#6 E6 B6 E6 E7 E6 B6 E6 G#6 E6 B6 E6 E7 E6 B6 E6},{2,E4 D5 D5 D5}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,G#6 E6 B6 E6 E7 E6 B6 E6 G#6 E6 B6 E6 E7 E6 B6 E6},{3/8,D5}{1/8,C5}{3/8,B4}{1/8,A4}{3/8,G#4}{1/8,A4}{3/8,B4}{1/8,G#4}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 E6 C7 E6 E7 E6 C7 E6 A6 E6 C7 E6 E7 E6 C7 E6},{1/8,B4}{1/4,A4}{1/8,G#4}{3/8,A4}{1/8,B4}{1/8,D5}{1/4,C5}{1/8,B4}{3/8,C5}{1/8,D5}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 E6 C7 E6 E7 E6 C7 E6 A6 E6 C7 E6 E7 E6 C7 E6},{3/8,D#5}{1/8,E5}{3/8,F5}{1/8,E5}{3/8,D5}{1/8,C5}{3/8,B4}{1/8,A4}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 E6 C#7 E6 E7 E6 C#7 E6 A6 E6 C#7 E6 E7 E6 C#7 E6},{2,E4 D5 D5 D5}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 E6 C#7 E6 E7 E6 C#7 E6 A6 E6 C#7 E6 E7 E6 C#7 E6},{3/8,G4}{1/8,F4}{3/8,E4}{1/8,D4}{3/8,C#4}{1/8,D4}{3/8,E4}{1/8,C#4}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 F6 D7 F6 F7 F6 D7 F6 A6 F6 D7 F6 F7 F6 D7 F6},{1/8,E4}{1/4,D4}{1/8,C#4}{3/8,D4}{1/8,E4}{1/8,G4}{1/4,F4}{1/8,E4}{3/8,F4}{1/8,G4}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 F6 D7 F6 F7 F6 D7 F6 A6 F6 D7 F6 F7 F6 D7 F6},{3/8,G#4}{1/8,A4}{3/8,Bb4}{1/8,A4}{3/8,G4}{1/8,F4}{3/8,E4}{1/8,D4}}}{_tempo(8/5) _volume(82) _chan(1){2,{2,A6 F6 D7 F6 F7 F6 D7 F6 A6 F6 D7 F6 F7 F6 D7 F6},{2,F3 Eb4 Eb4 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Cb5 D5 Bb4 Db5 A4 C5 Ab4 B4 G4 Bb4 Gb4 A4 F4 Ab4 E4 G4 D4 F4 Db4} 3/4 , 37/16 }}{_tempo(1/3) _volume(82) _chan(1){0/480,}}{_tempo(51/20) _volume(64) _chan(1){4,{2,C4 Bb4 Bb4 A4 A4 G4 G4 G#4}2,{1,-C3}{1/2,Bb3,E4}{2,C3 F4 --}{1/2,-C5}, 1/8 3/8 {1/2,A3,C4}{1,C3 -}}}{_tempo(51/20) _volume(64) _chan(1){2,{1,A4 C4}A4,{1/2,Bb3,F4}{1/2,C3}{1/2,Bb3,F4}{1/2,C3}}}{_tempo(51/20) _volume(64) _chan(1){2,{1,G4 C4}G4,{1/2,Bb3,E4}{1/2,C3}{1/2,Bb3,E4}{1/2,C3}}}{_tempo(51/20) _volume(64) _chan(1){2,{1/2,F4}{1/3,C4}{1/6,F4 G4}{1/2,F4}{1/2,E4 D4},{1/2,A3,C4}{1/2,C3}{1/2,Ab3,B3}{1/2,C3}}}{_tempo(51/20) _volume(64) _chan(1){2,{2,C4 Bb4 Bb4 A4 A4 G4 G4 G#4},{1/2,Bb3,E4}{1/2,C3}{1/2,Bb3,E4}{1/2,C3}}}{_tempo(51/20) _volume(64) _chan(1){2,{1,A4 C4}A4,{1/2,Bb3,F4}{1/2,C3}{1/2,Bb3,F4}{1/2,C3}}}{_tempo(51/20) _volume(64) _chan(1){2,{1,A4 C4}G4,{1/2,Bb3,E4}{1/2,C3}{1/2,Bb3,E4}{1/2,C3}}}{_tempo(51/20) _volume(64) _chan(1){2,{2,C5 Bb5 Bb5 A5 A5 G5 G5 G#5},C5 C5,{1,-C3}{1/2,Bb3,E4}{1/2,C3}}}{_tempo(51/20) _volume(64) 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_chan(1){2,{3/4,C#6}{1/4,A5}{3/4,C#6}{1/4,A5},{3/2,C#5 -C#5}1/2,{1/2,A2}{1/2,A3 C#4 E4}{1,A4 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,D6}{1/4,A5}{3/4,A6}{1/4,A5},{3/2,D5 -A5}1/2,{1/2,D3}{1/2,A3 D4 F#4}{1,A4 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{2,F6 E6 D6 C#6 D6 E6 F6 D6},{1/2,F5}3/2,{1/2,D3}{1/2,A3 D4 F4}{1,A4 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,E6}{1/4,A5}{3/4,E6}{1/4,D6},{3/2,E5 -E5}1/2,{1/2,A2}{1/2,A3 C#4 E4}{1,A4 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,C#6}{1/4,A5}{3/4,C#6}{1/4,A5},{3/2,C#5 -C#5}1/2,{1/2,A2}{1/2,A3 C#4 E4}{1,A4 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{D5,D6} 1/2 {1/2,-A5},{1/2,D3}{1/2,A3 D4 F#4}{1,A4 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{2,F7 E7 D7 C#7 D7 E7 F7 D7},{1/2,F6}3/2,{1/2,D4}{1/2,A4 D5 F5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,E7}{1/4,A6}{3/4,E7}{1/4,D7},{3/2,E6 -E6}1/2,{1/2,A3}{1/2,A4 C#5 E5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,C#7}{1/4,A6}{3/4,C#7}{1/4,A6},{3/2,C#6 -C#6}1/2,{1/2,A3}{1/2,A4 C#5 E5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,D7}{1/4,A6}{3/4,F#7}{1/4,A6},{3/2,D6 -F#6}1/2,{1/2,D4}{1/2,A4 D5 F#5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{2,F7 E7 D7 C#7 D7 E7 F7 D7},{1/2,F6}3/2,{1/2,D4}{1/2,A4 D5 F5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,E7}{1/4,A6}{3/4,E7}{1/4,D7},{3/2,E6 -E6}1/2,{1/2,A3}{1/2,A4 C#5 E5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,C#7}{1/4,A6}{3/4,C#7}{1/4,A6},{3/2,C#6 -C#6}1/2,{1/2,A3}{1/2,A4 C#5 E5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{1/4,D6,D7}{3/4,Bb6 A6 Bb6}A6,{1/2,D4}{1/2,A4 D5 F5}{1,A5 -}}}{_tempo(47/60) _volume(64) _chan(1){31/15,{2/3,A6}{1/3,G#6 A6 Bb6 D7}{16/15,F7 E7 D7 C#7 D7 C#7 Bb6 A6 Bb6 A6 G#6 F6 A6 G#6 F6 E6 G#6 F6 E6 D6 F6 E6 D6 C#6 D6 C#6 Bb5 A5 Bb5 A5 G#5 F5 A5 G#5 F5 E5 G#5 F5 E5 D5 F5 E5 D5 C#5 D5 C#5 Bb4 A4 Bb4 A4 G#4 F4 A4 G#4 F4 E4 G#4 F4 E4 D4 F4 E4 D4 C4}, 1/8 15/8 }}{_tempo(51/20) _volume(64) _chan(1){4,{2,C4 Bb4 Bb4 A4 A4 G4 G4 G#4}2,{1,-C3}{1/2,Bb3,E4}{2,C3 F4 --}{1/2,-C5}, 1/8 3/8 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_chan(1){2,{3/4,E7}{1/4,A6}{3/4,E7}{1/4,D7},{3/2,E6 -E6}1/2,{1/2,A3}{1/2,A4 C#5 E5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{3/4,C#7}{1/4,A6}{3/4,C#7}{1/4,A6},{3/2,C#6 -C#6}1/2,{1/2,A3}{1/2,A4 C#5 E5}{1,A5 -}}}{_tempo(7/3) _volume(64) _chan(1){2,{1/4,D6,D7}{3/4,Bb6 A6 Bb6}A6,{1/2,D4}{1/2,A4 D5 F5}{1,A5 -}}}{_tempo(47/60) _volume(64) _chan(1){31/15,{2/3,A6}{1/3,G#6 A6 Bb6 D7}{16/15,F7 E7 D7 C#7 D7 C#7 Bb6 A6 Bb6 A6 G#6 F6 A6 G#6 F6 E6 G#6 F6 E6 D6 F6 E6 D6 C#6 D6 C#6 Bb5 A5 Bb5 A5 G#5 F5 A5 G#5 F5 E5 G#5 F5 E5 D5 F5 E5 D5 C#5 D5 C#5 Bb4 A4 Bb4 A4 G#4 F4 A4 G#4 F4 E4 G#4 F4 E4 D4 F4 E4 D4 C4}, 1/8 15/8 }}{_tempo(169/60) _volume(64) _chan(1){2,{D6,D7} 1/2 {1/2,A5 Bb5 C#6 D6},{1/2,C4}{1/2,A4 D5 F5}{1,A5 -}}}{_tempo(169/60) _volume(113) _chan(1){2,{3/4,E5,E6}{1/4,F#5,F#6}{3/4,E5,E6}{1/4,F#5,F#6},{1/2,E2,E3}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}}}{_tempo(169/60) _volume(113) _chan(1){2,{3/4,E5,E6}{1/4,F#5,F#6}{1/3,E5,E6}{1/3,F#5,F#6}{1/3,E5,E6},{1/2,E2,E3}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}}}{_tempo(169/60) _volume(113) _chan(1){2,{1/3,F#5,F#6}{1/3,E5,E6}{1/3,F#5,F#6}{1/3,E5,E6}{1/3,F#5,F#6}{1/3,E5,E6},{1/2,E2,E3}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}}}{_tempo(169/60) _volume(113) _chan(1){2,{1/3,F#5,F#6}{1/3,E5,E6}{1/3,F#5,F#6}{1/3,G5,G6}{1/3,F#5,F#6}{1/3,E5,E6},{1/2,E2,E3}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}{1/2,A3,C#4,E4}}}{_tempo(169/60) _volume(113) _chan(1){2,{3/4,D5,D6}{1/4,F#5,A5}{3/4,D5,D6}{1/4,F#5,A5},{1/2,D2,D3}{1/2,A3,D4,F#4}{1/2,A3,D4,F#4}{1/2,A3,D4,F#4}}}{_tempo(169/60) _volume(113) _chan(1){2,{3/4,D5,D6}{1/4,F#5,A5}{1/4,D6}{1/4,D5,A5}{1/4,D6}{1/4,D5,A5},{1/2,D2,D3}{1/2,A3,D4,F#4}{1/2,A3,D4,F#4}{1/2,A3,D4,F#4}}}{_tempo(169/60) _volume(113) _chan(1){2,{3/4,D6}{1/4,F#5,A5}{1/4,D6}{1/4,D5,A5}{1/4,D6}{1/4,D5,A5},{1/2,D2,D3}{1/2,A3,D4,F#4}{1/2,A3,D4,F#4}{1/2,A3,D4,F#4}}}{_tempo(169/60) _volume(113) 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G#5 G#6},{1/2,C2,C3}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}}}{_tempo(11/4) _volume(113) _chan(1){2,{1/2,A5,A6}{1/2,C5,C6}{A5,A6},{1/2,C2,C3}{1/2,F3,A3,C4,F4}{1/2,F3,A3,C4,F4}{1/2,F3,A3,C4,F4}}}{_tempo(11/4) _volume(113) _chan(1){2,{1/2,G5,G6}{1/2,C5,C6}{G5,G6},{1/2,C2,C3}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}}}{_tempo(11/4) _volume(113) _chan(1){2,{1/2,F5,F6}{1/2,C5,C6}{Db5,Db6},{1/2,C2,C3}{1/2,F3,A3,C4,F4}{1/2,Ab3,B3,F4}{1/2,Ab3,B3,F4}}}{_tempo(11/4) _volume(113) _chan(1){2,{1/2,C5,C6}{3/2,Bb4 Bb5 A4 A5 G#4 G#5},{1/2,C2,C3}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}}}{_tempo(11/4) _volume(113) _chan(1){2,{1/2,A5,A6}{1/2,C5,C6}{A5,A6},{1/2,C2,C3}{1/2,F3,A3,C4,F4}{1/2,F3,A3,C4,F4}{1/2,F3,A3,C4,F4}}}{_tempo(11/4) _volume(113) _chan(1){2,{1/2,G5,G6}{1/2,C5,C6}{G5,G6},{1/2,C2,C3}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}{1/2,E3,Bb3,C4,E4}}}{_tempo(5/2) _volume(124) _chan(1){2,{1/2,F5,F6} 1/2 -,{1/2,F2,C3,F3}{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,F3,C4,F4}{1/4,G4,C5,G5}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,F3,C4,F4}{1/4,G4,C5,G5}{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,F3,C4,F4}{1/4,G4,C5,G5}}}{_tempo(5/2) _volume(124) _chan(1){2,{1/2,Db5,F5,Db6} 3/2 ‚{1/2,D2,A2,D3}{1/4,F3,Ab3,Db4}{1/4,Eb4,Ab4,Eb5}{1/4,F3,Ab3,Db4}{1/4,Eb4,Ab4,Eb5}{1/4,D3,Ab3,Db4}{1/4,E4,Ab4,Eb5}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,F3,Ab3,Db4}{1/4,Eb4,Ab4,Eb5}{1/4,Db3,Ab3,Db4}{1/4,Eb4,Ab4,Eb5}{1/4,F3,Ab3,Db4}{1/4,Eb4,Ab4,Eb5}{1/4,Db3,Ab3,Db4}{1/4,Eb4,Ab4,Eb5}}}{_tempo(5/2) _volume(124) _chan(1){2,{1/2,F5,A5,F6} 3/2 ‚{1/2,F2,C3,F3}{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,F3,C4,F4}{1/4,G4,C5,G5}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,F3,C4,F4}{1/4,G4,C5,G5}{1/4,A3,C4,F4}{1/4,G4,C5,G5}{1/4,F3,C4,F4}{1/4,Eb4,Eb5}}}{_tempo(5/2) _volume(124) _chan(1){2,{1/2,D5,F#5,D6} 3/2 ‚{1/2,D2,A2,D3}{1/4,F#3,A3,D4}{1/4,E4,A4,E5}{1/4,F#3,A3,D4}{1/4,E4,A4,E5}{1/4,D3,A3,D4}{1/4,E4,A4,E5}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,F3,A3,D4}{1/4,E4,A4,E5}{1/4,D3,A3,D4}{1/4,E4,A4,E5}{1/4,F3,A3,D4}{1/4,E4,A4,E5}{1/4,D3,A3,D4}{1/4,E4,A4,E5}}}{_tempo(5/2) _volume(124) _chan(1){2,{1/2,F#5,A#5,F#6} 3/2 ‚{1/2,F#2,C#3,F#3}{1/4,A#3,C#4,F#4}{1/4,G#4,C#5,G#5}{1/4,A#3,C#4,F#4}{1/4,G#4,C#5,G#5}{1/4,F#3,C#4,F#4}{1/4,G#4,C#5,G#5}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,A#3,C#4,F#4}{1/4,G#4,C#5,G#5}{1/4,F3,C#4,F#4}{1/4,G#4,C#5,G#5}{1/4,Db4,Gb4,Bb4}{1/4,C5,Gb5,C6}{1/4,Bb3,Gb4,Bb4}{1/4,C5,Gb5,C6}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,Gb4,Bb4,Db5}{1/4,Eb5,A5,Eb6}{1/4,Db4,Bb4,Db5}{1/4,Eb5,A5,Eb6}{1/4,Bb4,Db5,G5}{1/4,Ab5,Db6,Ab6}{1/4,Gb4,Db5,G5}{1/4,Ab5,Db6,Ab6}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,Db5,Gb5,Bb5}{1/4,C6,Gb6,C7}{1/4,Bb4,Gb5,Bb5}{1/4,C6,Gb6,C7}{1/4,Gb5,Bb5,Db6}{1/4,Eb6,A6,Eb7}{1/4,Db5,Bb5,Db6}{1/4,Eb6,A6,Eb7}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}{1/4,Db5,Gb5,Bb5,Db6}{1/4,Eb6,Gb6,Bb6,Eb7}}}{_tempo(5/2) _volume(113) _chan(1){2,--,{1/2,Db5,Gb5,Bb5,Db6} 1/2 -}}{_tempo(5/2) _volume(52) _chan(1){2,- 1/2 {1/2,-Db5},--}}{_tempo(5/2) _volume(113) _chan(1){2,{2,Bb5 Ab5 Gb5 F5 Gb5 Ab5 Bb5 Gb5},{1/2,Bb3,Bb4}{1/2,Db4,Gb4}{1/2,Bb3,Bb4}{1/2,Db4,Gb4}}}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,F5}{1/2,-Db5}{1/2,Ab5}{1/2,-Gb5},{1/2,C4,A4}{1/2,Db4,F4}{1/2,C4,A4}{1/2,Db4,F4}}}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,F5}{1/2,-Db5}{1/2,Db6}{1/2,-F5},{1/2,C4,A4}{1/2,Db4,F4}{1/2,C4,A4}{1/2,Db4,F4}}}{_tempo(5/2) _volume(113) _chan(1){2,Gb5 1/2 {1/2,-Db5},{1/2,Bb3,Bb4}{1/2,Db4,Gb4}{1/2,Bb3,Bb4}{1/2,Db4,Gb4}}}{_tempo(5/2) _volume(113) _chan(1){2,{2,Bb5 Ab5 Gb5 F5 Gb5 Ab5 Gb5 F5},{1/2,Bb3,Bb4}{1/2,Db4,Gb4}{1/2,Bb3,Bb4}{1/2,Db4,Gb4}}}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,E5}{1/2,-C5}{1/2,G5}{1/2,-F5},{1/2,Bb3,G4}{1/2,C4,E4}{1/2,Bb3,G4}{1/2,C4,E4}}}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,E5}{1/2,-C5}{1/2,C6}{1/2,-E5},{1/2,Bb3,G4}{1/2,C4,E4}{1/2,G3,E4}{1/2,Bb3,C4}}}{_tempo(5/2) _volume(113) _chan(1){2,F5 1/2 {1/2,-C6},{1/2,F3,F4}{1/2,A3,C4}{1/2,Eb3,Eb4}{1/2,A3,C4},{2,F4 -Eb4 -}}}{_tempo(5/2) _volume(113) _chan(1){2,{2,Bb6 Ab6 Gb6 F6 Gb6 Ab6 Bb6 Gb6},{1/2,Eb3,Eb4}{1/2,Gb3,Bb3}{1/2,D3,D4}{1/2,Gb3,Bb3},{1/2,Eb4} 3/2 }}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,F6}{1/2,-Db6}{1/2,Ab6}{1/2,-Gb6},{1/2,Db3,Db4}{1/2,F3,Ab3}{1/2,Db3,Db4}{1/2,F3,Ab3}}}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,F6}{1/2,-Db6}{1/2,Db7}{1/2,-F6},{1/2,Db3,Db4}{1/2,F3,Ab3}{1/2,Cb3,Cb4}{1/2,F3,Ab3}}}{_tempo(5/2) _volume(113) _chan(1){2,Gb6 1/2 {1/2,-Db4},{1/2,Bb2,Bb3}{1/2,Db3,Gb3}{1/2,Bb2,Bb3}{1/2,Db3,Gb3}}}{_tempo(5/2) _volume(113) _chan(1){2,{2,Bb4 Ab4 Gb4 F4 Gb4 Ab4 Gb4 F4},{1/2,Bb2,Bb3}{1/2,Db3,Gb3}{1/2,Bb2,Bb3}{1/2,Db3,Gb3}}}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,E4}{1/2,-C4}{1/2,G4}{1/2,-F4},{1/2,Bb2,G3}{1/2,Db3,E3}{1/2,A2,G3}{1/2,C#3,E3},-{1,G3 -}}}{_tempo(5/2) _volume(113) _chan(1){2,{1/2,E4}{1/2,-Db4}{1/2,C5}{1/2,-E4},{1/2,G#2,G#3}{1/2,C#3,E3}{1/2,G2,Bb3}{1/2,C3,E3},{2,G#3 -Bb3 -}}}{_tempo(179/60) _volume(97) _chan(1){2,{1/2,F4}{1/2,A3,C4}{1/2,A4}{1/2,A3,C4},{1/2,F2}{1/2,C3,F3}{1/2,Eb2}{1/2,C3,Eb3}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,F#4}{1/2,A3,C4}{1/2,D5}{1/2,D4,F#4,C5},{3/2,D2 Eb3 D3}{1/2,F#3,A3}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,B4}{1/2,D4,F4}{1/2,G5}{1/2,G4,B4,F5},{3/2,G2 Ab3 G3}{1/2,B3,D4}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,F#5}{1/2,A4,C5}{1/2,D6}{1/2,D5,F#5,C6},{3/2,C3 Db4 C4}{1/2,E4,G4}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,A5}{1/2,A4,C5}{1/2,A5}{1/2,A4,C5,G5},{1/2,F3}{1/2,C4,F4}{1/2,Eb3}{1/2,C4,Eb4}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,F#5}{1/2,A4,C5}{1/2,D6}{1/2,D5,F#5,C6},{3/2,D3 E4 D4}{1/2,F#4,A4}}}{_tempo(179/60) _volume(97) _chan(1){2,{1/2,B5}{1/2,D5,F5}{1/2,G6}{1/2,G5,B5,F6},{3/2,G3 Ab4 G4}{1/2,B4,D5}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,E6}{1/2,G5,Bb5}{1/2,C7}{1/2,C6,E6,Bb6},{3/2,C4 Db5 C5}{1/2,E5,G5}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,A6}{1/2,A5,D6}{1/2,A6}{1/2,A5,D6},{3/2,A6 -A6}1/2,{1/2,F4}{1/2,C5,F5}{1/2,D4}{1/2,D5,F#5},-{1,D4 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,Bb6}{1/2,Bb5,D6}{1/2,B6}{1/2,B5,E6},{3/2,Bb6 -B6}1/2,{1/2,G4}{1/2,D5,G5}{1/2,E4}{1/2,E5,G#5},{2,G4 -E4 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,C7}{1/2,C6,E6}{1/2,C#7}{1/2,C#6,E6},{3/2,C7 -C#7}1/2,{1/2,A4}{1/2,E5,A5}{1/2,A4}{1/2,C#5,E5,A5},{2,A4 -A4 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,D7}{1/2,D6,F6}{1/2,E7}{1/2,D6,F6},{3/2,D7 -E7}1/2,{1/2,Bb3}{1/2,D4,F4,Bb4}{1/2,A3}{1/2,E4,A4},{2,Bb3 -A3 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,F7}{1/2,F6,A6}{1/2,A6}{1/2,A5,D6},{3/2,F7 -A6}1/2,{1/2,F4}{1/2,A4,C5,F5}{1/2,D4}{1/2,F#4,A4,D5},{2,F4 -D4 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,Bb6}{1/2,Bb5,D6}{1/2,B6}{1/2,B5,E6},{3/2,Bb6 -B6}1/2,{1/2,E3}{1/2,G3,Bb3,E4}{1/2,E3}{1/2,G#3,B3,E4},{2,E3 -E3 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,C7}{1/2,C6,E6}{1/2,C#7}{1/2,C#6,E6},{3/2,C7 -C#7}1/2,{1/2,A3}{1/2,C4,E4,A4}{1/2,A3}{1/2,C#4,E4,A4},{2,A3 -A3 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,D7}{1/2,D6,F6}{1/2,E7}{1/2,E6,G6},{3/2,D7 -E7}1/2,{1/2,D3}{1/2,F3,A3,D4}{1/2,D3}{1/2,G3,C4},{2,D3 -D3 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,F7}{1/2,F6,A6}{1/2,A5,A6}{1/2,D6,F#6},{3/2,F7 -A6}1/2,{1/2,F3}{1/2,A3,C4,F4}{1/2,D3,D4}{1/2,F#3,A3},{2,F3 -D4 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,Bb5,Bb6}{1/2,D6,G6}{1/2,B5,B6}{1/2,E6,G#6},{3/2,Bb6 -B6}1/2,{1/2,G2,G3}{1/2,Bb2,D3}{1/2,E2,E3}{1/2,G#2,B2},{2,G3 -E3 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,C6,C7}{1/2,E6,A6}{1/2,C#6,C#7}{1/2,E6,A6},{3/2,C7 -C#7}1/2,{1/2,A2,A3}{1/2,C3,E3}{1/2,A2,A3}{1/2,C#3,E3},{2,A3 -A3 -}}}{_tempo(179/60) _volume(113) _chan(1){2,{1/2,D6,D7}{1/2,F6,A6}{1/2,E6,E7}{1/2,G6,C7},{3/2,D7 -E7}1/2,{1/2,D2,D3}{1/2,F2,A2}{1/2,C2,G2,C3} 1/2 ‚{1/2,D3} 3/2 }}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,F6,A6,C7,F7}{1/2,F6,F7}{1/2,E6,E7}{1/2,D6,D7},{1/2,F2,A2,C3,F3}{1/2,F2,F3}{1/2,E2,E3}{1/2,D2,D3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,C6,C7}{1/4,D6,D7}{1/4,E6,E7}{1/2,F6,F7}{1/2,A6,A7},{1/2,C2,C3}{1/4,D2,D3}{1/4,E2,E3}{1/2,F2,F3}{1/2,A2,A3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,G6,G7}{1/2,B4,D5,G5,B5}{C5,E5,G5,C6},{1/2,G2,G3}{1/2,G3,B3,D4,G4}{C3,E3,G3,C4}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,Bb5,Bb6}{1/2,Bb5,Bb6}{1/2,A5,A6}{1/2,G5,G6},{1/2,Bb2,Bb3}{1/2,Bb2,Bb3}{1/2,A2,A3}{1/2,G2,G3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,C6,C7}{1/4,D6,D7}{1/4,E6,E7}{1/2,F6,F7}{1/2,A6,A7},{1/2,F2,F3}{1/4,G2,G3}{1/4,A2,A3}{1/2,Bb2,Bb3}{1/2,D3,D4}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,C6,C7}{1/2,E5,G5,C6,E6}{F5,A5,C6,F6},{1/2,C3,C4}{1/2,C4,E4,G4,C5}{F3,A3,C4,F4}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,F3,F4}{1/2,F3,F4}{1/2,E3,E4}{1/2,D3,D4},{1/2,F1,F2}{1/2,F1,F2}{1/2,E1,E2}{1/2,D1,D2}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,C3,C4}{1/4,D3,D4}{1/4,E3,E4}{1/2,F3,F4}{1/2,A3,A4},{1/2,C1,C2}{1/4,D1,D2}{1/4,E1,E2}{1/2,F1,F2}{1/2,A1,A2}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,G3,G4}{1/2,B4,D5,G5,B5}{C5,E5,G5,C6},{1/2,G1,G2}{1/2,G3,B3,D4,G4}{C3,E3,G3,C4}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,Bb3,Bb4}{1/2,Bb3,Bb4}{1/2,A3,A4}{1/2,G3,G4},{1/2,Bb1,Bb2}{1/2,Bb1,Bb2}{1/2,A1,A2}{1/2,G1,G2}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,C4,C5}{1/4,D4,D5}{1/4,E4,E5}{1/2,F4,F5}{1/2,A4,A5},{1/2,F1,F2}{1/4,G1,G2}{1/4,A1,A2}{1/2,Bb1,Bb2}{1/2,D2,D3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,C4,C5}{1/2,E4,G4,C5,E5}{1/2,F4,A4,C5,F5}{1/2,G4,C5,E5,G5},{1/2,C2,C3}{1/2,C3,E3,G3,C4}{1/2,F2,A2,C3,F3}{1/2,C2,E2,G2,C3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,A4,C5,F5,A5}{1/2,C5,E5,G5,C6}{1/2,C5,F5,A5,C6}{1/2,E4,G4,C5,E5},{1/2,F2,A2,C3,F3}{1/2,C3,E3,G3,C4}{1/2,F2,A2,C3,F3}{1/2,C2,E2,G2,C3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,F4,A4,C5,F5}{1/2,G4,C5,E5,G5}{1/2,A4,C5,F5,A5}{1/2,C5,E5,G5,C6},{1/2,F2,A2,C3,F3}{1/2,C3,E3,G3,C4}{1/2,F2,A2,C3,F3}{1/2,C2,E2,G2,C3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,C5,F5,A5,C6}{1/2,E5,G5,C6,E6}{1/2,F5,A5,C6,F6}{1/2,E5,G5,C6,E6},{1/2,F2,A2,C3,F3}{1/2,C2,E2,G2,C3}{1/2,F2,A2,C3,F3}{1/2,C2,E2,G2,C3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,F5,A5,C6,F6}{1/2,E5,G5,C6,E6}{1/2,F5,A5,C6,F6}{1/2,E5,G5,C6,E6},{1/2,F2,A2,C3,F3}{1/2,C2,E2,G2,C3}{1/2,F2,A2,C3,F3}{1/2,C2,E2,G2,C3}}}{_tempo(7/3) _volume(113) _chan(1){9/4,{1/2,F5,A5,C6,F6} 1/4 {1/4,A4,C5,F5,A5}{1/2,A4,C5,F5,A5} 1/4 {1/4,C5,F5,A5,C6} 1/4 ‚{1/2,F2,A2,C3,F3} 1/4 {1/4,F3,A3,C4,F4}{1/2,F3,A3,C4,F4} 1/4 {1/4,C3,F3,A3,C4} 1/4 }}{_tempo(7/3) _volume(113) _chan(1){9/4, 1/4 {1/2,C5,F5,A5,C6} 1/4 {1/4,F4,A4,C5,F5}{1/2,F4,A4,C5,F5} 1/4 {1/4,A4,C5,F5,A5}, 1/4 {1/2,C3,F3,A3,C4} 1/4 {1/4,A2,C3,F3,A3}{1/2,A2,C3,F3,A3} 1/4 {1/4,F2,A2,C3,F3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,A4,C5,F5,A5} 1/4 {1/4,F4,A4,C5,F5}{1/2,F4,A4,C5,F5} 1/4 {1/4,A4,C5,F5,A5},{1/2,F2,A2,C3,F3} 1/4 {1/4,A2,C3,F3,A3}{1/2,A2,C3,F3,A3} 1/4 {1/4,F2,A2,C3,F3}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,A4,C5,F5,A5} 1/4 {1/4,C5,F5,A5,C6}{1/2,C5,F5,A5,C6} 1/4 {1/4,F5,A5,C6,F6},{1/2,F2,A2,C3,F3} 1/4 {1/4,C2,F2,A2,C3}{1/2,C2,F2,A2,C3} 1/4 {1/4,F1,A1,C2,F2}}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,F5,A5,C6,F6} 1/2 -,{1/2,F1,A1,C2,F2} 1/2 -}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,A4,C5,F5,A5} 1/2 -,{1/2,C3,F3,A3,C4} 1/2 -}}{_tempo(7/3) _volume(113) _chan(1){2,{1/2,F3,A3,C4,F4} 1/2 -,{1/2,F1,A1,C2,F2} 1/2 -}}

Return to humanity

Examples will hope­ful­ly con­vince read­ers that the Bol Processor for­mat is able to emu­late detailed scores in com­mon Western music nota­tion, and even fix some irreg­u­lar­i­ties in their tim­ings… Let us admit a long way from its ini­tial ded­i­ca­tion to the beau­ti­ful poet­ry cre­at­ed by drum play­ers in India!

These are indeed inter­pre­ta­tions of musi­cal scores. In order to remem­ber the addi­tion­al val­ue cre­at­ed by human artists play­ing real instru­ments, we may end up lis­ten­ing to the same Beethoven’s Fugue played by Alban Berg Quartett:

Beethoven’s Fugue in B flat major (opus 133). Source: https://youtu.be/13ygvpIg-S0

A multicultural model of consonance

A frame­work for tun­ing just-intonation scales via two series of fifths

For more than twen­ty cen­turies, musi­cians, instru­ment mak­ers and musi­col­o­gists fig­ured out scale mod­els and tun­ing pro­ce­dures for the cre­ation of music embody­ing the con­cept of “con­so­nance”. There was a shared notion of the octave and the major fifth (inter­val “C” to “G”) being the build­ing blocks of these mod­els, and the har­mon­ic major third (inter­val “C” to “E”) late­ly played a sig­nif­i­cant role in European baroque and clas­si­cal music.

Computer-controlled elec­tron­ic instru­ments open new avenues for the imple­men­ta­tion of micro­tonal­i­ty includ­ing just-intonation frame­works divid­ing the octave in more than 12 grades (https://bolprocessor.org/microtonality/). Throughout cen­turies, Indian art music claimed its adher­ence to a divi­sion of 22 inter­vals (the ṣruti-swara sys­tem) the­o­rized in Nāṭyaśāstra, a Sanskrit trea­tise dat­ing back between 400 BCE to 200 CE. Since con­so­nance (saṃvādī) is the basis of both ancient Indian and European tonal sys­tems, we felt the urge for a the­o­ret­i­cal frame­work encom­pass­ing all models.

Unfortunately, the top­ic of “just into­na­tion” is exposed in an alto­geth­er con­fus­ing and reduc­tive man­ner (read Wikipedia) due to musi­col­o­gists’ focus on inte­ger ratios aimed at reflect­ing the dis­tri­b­u­tion of high­er par­tials in peri­od­i­cal sounds. While these spec­u­la­tive mod­els of into­na­tion may respond to beliefs in the mys­ti­cal prop­er­ties of nat­ur­al num­bers — as claimed by Pythagoreanists — they were rarely checked against non-directed musi­cal prac­tice. Instrument tuners most­ly rely on their own audi­to­ry per­cep­tion of inter­vals rather than resort­ing to num­bers, despite the avail­abil­i­ty of “elec­tron­ic tuners”…

Interestingly, the ancient Indian the­o­ry of nat­ur­al scales does not rely on arith­metics. This should be sur­pris­ing giv­en that in Vedic times mathematicians/philosophers had laid out the foun­da­tions of cal­cu­lus and infin­i­tes­i­mals which much lat­er were export­ed from Kerala to Europe and borrowed/appropriated by European schol­ars — read C.K. Raju’s Cultural Foundations of Mathematics: the nature of math­e­mat­i­cal proof and the trans­mis­sion of the cal­cu­lus from India to Europe in the 16th c. CE. This epis­te­mo­log­i­cal para­dox was an incen­tive to decrypt the mod­el depict­ed by the author(s) of Nāṭyaśāstra via a thought exper­i­ment: the two-vina exper­i­ment (https://bolprocessor.org/two-vina-experiment).

Earlier inter­pre­ta­tions of this mod­el, mim­ic­k­ing the Western habit of deal­ing with inter­vals as fre­quen­cy ratios, failed to explain the inter­val­ic struc­ture of ragas in Hindustani clas­si­cal music. In fact, the implic­it mod­el of Nāṭyaśāstra is a “flex­i­ble” one because the size of the major third (or equiv­a­lent­ly the pramāņa ṣru­ti) is not stat­ed in advance. Read Raga into­na­tion (https://bolprocessor.org/raga-intonation) and lis­ten to exam­ples to grasp the artic­u­la­tion between the the­o­ry and prac­tice of into­na­tion in this context.

In Europe, the har­mon­ic major third was final­ly accept­ed as a “celes­tial inter­val” after the Council of Trent (1545-1563) putting an end to the ban­ish­ment of poly­phon­ic singing in reli­gious gath­er­ings. Along with the devel­op­ment of fixed-pitch key­board instru­ments, this gave way to the elab­o­ra­tion of the­o­ret­i­cal mod­els and tun­ing pro­ce­dures attempt­ing to include this inter­val in “pure into­na­tion”. In the­o­ry, this is not fea­si­ble on a chro­mat­ic (12-grade scale) but it can be fig­ured out and applied to Western har­mo­ny if more grades (29 to 41) are per­mit­ted. However, choos­ing enhar­mon­ic posi­tions suit­able for a har­mon­ic con­text remains an uncer­tain venture.

Once again, the Indian mod­el came to the res­cue because it can be extend­ed to pro­duce a con­sis­tent series of twelve “opti­mal­ly con­so­nant” chro­mat­ic scales in com­pli­ance with chord inter­vals in Western har­mo­ny. Each scale con­tains 12 grades, which is more than the notes of chords it is applic­a­ble for. Sound exam­ples are pro­vid­ed to illus­trate this process (https://bolprocessor.org/just-intonation-framework/).

Tuning mechan­i­cal key­board instru­ments (church organ, harp­si­chord, pianoforte) for 12-grade scales made it nec­es­sary to dis­trib­ute unwant­ed dis­so­nance (the syn­ton­ic com­ma) in an accept­able man­ner over series of fifths and fourths. Many tem­pered tun­ing pro­ce­dures were designed, dur­ing the 16th to 19th cen­turies, with empha­sis on either “per­fect fifths” or “pure major thirds”, in response to the con­straints of spe­cif­ic musi­cal repertoires.

These tech­niques have been doc­u­ment­ed in detail by organ play­er and instru­ment design­er Pierre-Yves Asselin, along with meth­ods for achiev­ing the tun­ing on a mechan­i­cal instru­ment such as the harp­si­chord. His book Musique et tem­péra­ment (soon avail­able in English) was a guide­line for imple­ment­ing a sim­i­lar approach in the Bol Processor (https://bolprocessor.org/microtonality/). This frame­work will make it pos­si­ble to lis­ten to baroque and clas­si­cal works, using Csound instru­ments, with the very tun­ings their com­posers had been favor­ing — accord­ing to his­tor­i­cal sources.

Creation of just-intonation scales

The fol­low­ing is the pro­ce­dure for export­ing just-intonation scales from mur­ccha­na-s of Ma-grama stored in “-cs.12_scales”.

Read Just into­na­tion: a gen­er­al frame­work for explanations.

The scale model

From left to right: 1st-order descending-third series, “Pythagorean” series and 1st-order ascending-third series (Asselin 2000 p. 61)

As indi­cat­ed on page Just into­na­tion: a gen­er­al frame­work, just-intonation chro­mat­ic scales can be derived from a basic frame­work made of two cycles of per­fect fifths (fre­quen­cy ratio 3/2).

These pro­duce the 22-shru­ti frame­work of Indian musi­col­o­gists (read Raga into­na­tion) or the series named “Pythagorean” and “1st-order ascending-third” (“LA-1”, “MI-1” etc.) in the approach of Western musi­col­o­gists (see pic­ture on the side).

We found that the “1st-order descending-third cycle” (“LAb+1”, “MIb+1” etc.) in which all notes are high­er by a syn­ton­ic com­ma may not be required for the cre­ation of just-intonation chords.

These cycles of fifths are rep­re­sent­ed graph­i­cal­ly (scale “2_cycles_of_fifths” in Csound resource “-cs.tryTunings”):

There are a few dif­fer­ences between this 29-grade divi­sion of the octave and the Indian frame­work, notably the cre­ation of “DO-1” and “FA-1”, two posi­tions low­er by one syn­ton­ic com­ma than “DO” (“C” = “Sa” in the Indian con­ven­tion) and “FA” (“F” = “Ma”). Interestingly, these posi­tions appear in ancient texts under the names “cyu­ta Sa” and “cyu­ta Ma”. Other addi­tion­al posi­tions are “REb-1”, “MIb-1”, “SOLb-1”, “LAb-1” and “SIb-1”.

The rule we fol­low when pro­duc­ing chro­mat­ic scales via trans­po­si­tions of Ma-grama is that only posi­tions dis­played on this graph should be con­sid­ered valid. When export­ing a minor or major chro­mat­ic scale from a trans­po­si­tion of Ma-grama, it may occur that a note posi­tion is not part of this frame­work. In all cas­es of this pro­ce­dure, the invalid posi­tion is one syn­ton­ic com­ma too low. Therefore the export­ed scale will be “aligned” rais­ing all its posi­tions by one comma.

The term “Pythagorean series” is con­fus­ing because any cycle of per­fect fifths is Pythagorean by def­i­n­i­tion. Whether a posi­tion in a scale “is” or “is not” Pythagorean depends on the start­ing note of the series that was announced as “Pythagorean”. In Asselin’s work the start­ing point of the series in the cen­tral col­umn is “FA”. In the Indian sys­tem, basic frame­works (Ma-grama and Sa-grama) start from “Sa” (“C” or “do”) and the Pythagorean/harmonic sta­tus of a posi­tion is deter­mined by fac­tors of its fre­quen­cy ratio with respect to “Sa”. If a fac­tor “5” is found in the numer­a­tor or the denom­i­na­tor, the posi­tion is har­mon­ic, or Pythagorean in the reverse case.

Thus, for instance, “DO#” in Asselin’s “Pythagorean” series (two per­fect fifths above “SI”) is eval­u­at­ed as a har­mon­ic posi­tion (marked in green) on the Bol Processor graph­ic and its ratio is 16/15. In real­i­ty, “DO#” in Asselin’s series has a fre­quen­cy ratio 243/128 * 9/16 = 2187/1024 = 1.068 which is very close to 16/15 = 1.067. “DO#-1” in Asselin’s series is two per­fect fifths above “SI-1” which yields a fre­quen­cy ratio 15/8 * 9/16 = 135/128 = 1.054 which is close to 256/243 = 1.053 and marked “Pythagorean” on the Indian scheme. Thus, “DO#” and “DO#-1” have exchanged their prop­er­ties because each of them is the super­po­si­tion of two very close posi­tions belong­ing to dif­fer­ent series.

Ignoring schis­ma dif­fer­ences to take the sim­plest ratios cre­ate this con­fu­sion. Therefore, we keep pre­fer­ring com­ma indi­ca­tions — e.g. “FA” and “FA-1” — to iden­ti­fy posi­tions, in which the first instance belongs to the series termed “Pythagorean” in Asselin’s work.

Transposition table

This table sum­ma­rizes a quick pro­ce­dure for cre­at­ing all mur­ccha­na-s of the Ma-grama chro­mat­ic scale and export minor and major chro­mat­ic scales therefrom.

Open the “Ma_grama” scale in the “-cs.12_scales” Csound resource and select the Murcchana pro­ce­dure. To cre­ate “Ma01″, move note “F” to note “C” and click TRANSPOSITION.

F moved toMurcchanaMinor scaleRaiseMajor scaleIdentical
scale
Adjust
CMa01AminDCmaj=Emin1/1
FMa02DminGFmaj=Amin1/1
BbMa03GminCBbmaj=Dmin1/1
EbMa04CminFEbmaj=Gmin1/1
AbMa05FminBbAbmaj=Cmin1/1
DbMa06BbminEbDbmaj=Fmin1/1
F#Ma07EbminAbF#maj=Bbmin1/1
BMa08AbminDbBmaj=Ebmin1/1
EMa09DbminF#Emaj=Abmin1/1
AMa10F#minBAmaj=Dbmin81/80
R3Ma11BminEDmaj=F#min81/80
G3Ma12EminAGmaj=Bmin81/80

For exam­ple, this is the “Ma04mur­ccha­na obtained by plac­ing “F” (M1 on the Indian scale mod­el) of the move­able wheel on “Eb” (G1 of the out­er crown):

The result­ing scale “Ma04″ is:

The “Ma04” scale pro­duced as a trans­po­si­tion of the “Ma-grama” chro­mat­ic scale

Scale adjustment

In the last col­umn of the table, “Adjust” indi­cates the frac­tion by which the ratios of notes may need to be mul­ti­plied so that no posi­tion is cre­at­ed out­side the Pythagorean and har­mon­ic cycles of fifths accord­ing to the Indian sys­tem. Practically this is the case when the fre­quen­cy ratio con­tains a mul­ti­ple of 25 in either its numer­a­tor or denom­i­na­tor, as this indi­cates that the posi­tion has been con­struct­ed by at least two suc­ces­sive major thirds (up or down). 

A warn­ing is dis­played when this is the case, and a sin­ple click on ADJUST SCALE fix­es positions:

In this exam­ple, the warn­ing sig­nals an out-of-range posi­tion of “B” (50/27) on the “Ma10″ scale. Also note that “F#” has a mul­ti­ple of 25 in its numerator.

After click­ing ADJUST SCALE, the “Ma10″ scale is final­ized with “B” at posi­tion 15/8. This has been done by rais­ing all notes by a syn­ton­ic com­ma (81/80) :

This pro­ce­dure is men­tioned in Indian musi­col­o­gy under the name of sadja-sadharana telling that all notes of the scale are raised by one shru­ti — here, a syn­ton­ic com­ma (Shringy & Sharma 1978). In this mod­el, it is also invoked for scales “Ma11″ and “Ma12″. The result is (as expect­ed) a cir­cu­lar mod­el because “Ma13″ is iden­ti­cal to “Ma01″ as shown by the scale com­para­tor at the bot­tom of page “-cs.12_scales”.

This cir­cu­lar­i­ty is a prop­er­ty of the set of mur­ccha­na-s which has no influ­ence on export­ed minor and major scales because their posi­tions will be aligned in com­pli­ance with the basic rule exposed in the first section.

Exporting and aligning minor scales

The “Ma04mur­ccha­na pro­duces “Cmin” by export­ing notes fac­ing marks on the inner wheel.

The “Cmin” chro­mat­ic scale export­ed from the “Ma04” transposition

As explained on page Just into­na­tion: a gen­er­al frame­work, the ton­ic and dom­i­nant notes of every minor chord should belong to the “minus-1” posi­tion. In this exam­ple, “C” and “G” are one com­ma low­er in a “C minor” chord than in a “C major” chord (match­ing “DO-1” and “SOL-1” on the “2_cycles_of_fifths” scale) , a fact which had been pre­dict­ed and exper­i­men­tal­ly checked by Pierre-Yves Asselin (2000 p. 137).

All chro­mat­ic minor scales export­ed from mur­chana-s of Ma-grama are cor­rect­ly posi­tioned with respect of enhar­mon­ic posi­tions of main notes in just-intonation chords. This can eas­i­ly be checked com­par­ing ratios with the ones asso­ci­at­ed with the Western series on “2_cycles_of_fifths” (top of this page). This con­firms that a tun­ing sys­tem using only two series of per­fect fifths is con­ve­nient for the con­struc­tion of a just-intonation framework.

Exporting and aligning major scales

The “Ma04mur­ccha­na pro­duces “Ebmaj” by export­ing notes fac­ing marks on the inner wheel and rais­ing “F”:

The “Ebmaj” chro­mat­ic scale export­ed from the “Ma04” transposition

According to a rule exposed on page Just into­na­tion: a gen­er­al frame­work, the basic note of every major chord should be both in the high posi­tion and in the Pythagorean series (blue mark­ings). This is true of “Eb major” chord extract­ed from the “Ebmaj” chro­mat­ic scale, yet not with scales “F#maj”, “Bmaj” and “Emaj” dis­played in bold style on the table.

Let us for instance look at “Emaj” export­ed with­out pre­cau­tion from “Ma09″:

Scale “Emaj” export­ed from “Ma09”, before its alignment

Note “E” has a fre­quen­cy ratio 5/4 which is labeled “MI-1” on scale “2_cycles_of_fifths” (top of this page). Since “MI-1” belongs to a har­mon­ic series, it can­not be tak­en as a the ton­ic of a “E major chord”. The Pythagorean “MI” (ratio 81/64) should be used instead.

After its align­ment — rais­ing all notes by 1 syn­ton­ic com­ma — the final “Emaj” scale is obtained:

Scale “Emaj” export­ed from “Ma09”, after its alignment

This align­ment of export­ed major scales is done auto­mat­i­cal­ly by the Bol Processor when export­ing a major chro­mat­ic scale.

References

Asselin, P.-Y. Musique et tem­péra­ment. Paris, 1985, repub­lished in 2000: Jobert. Soon avail­able in English.

Shringy, R.K.; Sharma, P.L. Sangita Ratnakara of Sarngadeva: text and trans­la­tion, vol. 1, 5: 7-9. Banaras, 1978: Motilal Banarsidass. Source in the Web Archive.

Raga intonation

Tanpura: the drone of Indian musi­cians (man­u­fac­tured in Miraj)

This arti­cle demon­strates the the­o­ret­i­cal and prac­ti­cal con­struc­tion of micro­ton­al scales for the into­na­tion of North Indian ragas, using tools avail­able with Bol Processor (BP3) + Csound.

It comes as a com­ple­ment to pages Microtonality and Just into­na­tion, a gen­er­al frame­work and The Two-vina exper­i­ment. Nonetheless, its under­stand­ing does not require a pre­lim­i­nary study of these relat­ed pages.

This exer­cise on raga into­na­tion demon­strates the abil­i­ty of BP3 to deal with sophis­ti­cat­ed mod­els of micro-intonation and sup­port a fruit­ful cre­ation of music embod­ied by these models.

Theory versus practice

To sum­ma­rize the back­ground, the frame­work for con­struct­ing “just-intonation” scales is a deci­pher­ing of the first six chap­ters of Nāṭyaśāstra, a Sanskrit trea­tise on music, dance and dra­ma dat­ing back to a peri­od between 400 BCE and 200 CE. For con­ve­nience we call it “Bharata’s mod­el” although there is no his­tor­i­cal record of a sin­gle author bear­ing this name.

Using exclu­sive infor­ma­tion dri­ven from the text and its descrip­tion of the Two-vina exper­i­ment pro­vides an infi­nite set of valid inter­pre­ta­tions of the ancient the­o­ry as shown in A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra (Bel 1988). Among these, the one advo­cat­ed by many musi­col­o­gists — influ­enced by Western acoustics and scale the­o­ries — states that the fre­quen­cy ratio of the har­mon­ic major third would be 5/4. This is equiv­a­lent to fix­ing the fre­quen­cy ratio of the syn­ton­ic com­ma to 81/80.

Even though this inter­pre­ta­tion yields a con­sis­tent mod­el for just-intonation har­mo­ny — read Just into­na­tion, a gen­er­al frame­work — it would be far-fetched to stip­u­late that the same holds for raga into­na­tion. Accurate mea­sure­ments of raga per­for­mance using our Melodic Movement Analyzer in the ear­ly 1980s revealed that melod­ic struc­tures inferred from sta­tis­tics (using selec­tive tona­grams, read below) often dif­fer sig­nif­i­cant­ly from scales pre­dict­ed by the “just-intonation” inter­pre­ta­tion of Bharata’s mod­el. Part of the expla­na­tion may be the strong har­mon­ic attrac­tion effect of drones (tan­pu­ra) played in the back­ground of per­for­mances of raga.

Talking about gra­ma-s (scale frame­works) in the ancient Indian the­o­ry, E.J. Arnold wrote (1982 p. 40):

Strictly speak­ing the gra­mas belong to that aspect of nada (vibra­tion) which is ana­ha­ta (“unstruck”). That means to say that the “gra­ma” can nev­er be heard as a musi­cal scale [as we did on page Just into­na­tion, a gen­er­al frame­work]. What can be heard as a musi­cal scale is not the gra­ma, but any of its mur­ccha­nas.

As soon as elec­tron­ic devices such as the Shruti Harmonium (1979) and the Melodic Movement Analyzer (1981) became avail­able, the chal­lenge of research on raga into­na­tion was to rec­on­cile two method­olo­gies: a top-down approach check­ing hypo­thet­i­cal mod­els against data, and a data-driven bottom-up approach.

The “micro­scop­ic” obser­va­tion of melod­ic lines (now ren­dered easy by soft­ware like Praat) con­firmed the impor­tance of note treat­ment (orna­men­ta­tion, alankara) and time-driven dimen­sions of raga which are not tak­en into account by scale the­o­ries. For instance, long dis­cus­sions have been held on the ren­der­ing of note “Ga” in raga Darbari Kanada (Bel & Bor 1984; van der Meer 2019) and typ­i­cal treat­ment of notes in oth­er ragas (e.g. Rao & Van der Meer 2009; 2010). The visu­al tran­scrip­tion of a phrase of raga Asha makes it evident:

A brief phrase of raga Asha tran­scribed by the MMA and in Western con­ven­tion­al notation
Non-selective tona­gram of raga Sindhura sung by Ms. Bhupender Seetal

In order to extract scale infor­ma­tion from this melod­ic con­tin­u­um, a sta­tis­ti­cal mod­el was imple­ment­ed to dis­play the dis­tri­b­u­tion of pitch across one octave. The image shows the tona­gram of a 2-minute sketch (cha­lana) of raga Sindhura taught by Pandit Dilip Chandra Vedi.

The same record­ing of Sindhura on a selec­tive tonagram

The same melod­ic data was processed again after a fil­ter­ing of 3 win­dows attempt­ing to iso­late “sta­ble” parts of the line. The first win­dow, typ­i­cal­ly 0.1 sec­onds, would elim­i­nate irreg­u­lar seg­ments, the sec­ond one (0.4 s.) would dis­card seg­ments out­side a rec­tan­gle of 80 cents in height, and the third one was used for aver­ag­ing. The out­come is a “skele­ton” of the tonal scale dis­played as a selec­tive tona­gram.

These results often would not match scale met­rics pre­dict­ed by the “just-intonation” inter­pre­ta­tion of Bharata’s mod­el. Proceeding fur­ther in this data-driven approach, we pro­duced the (non-selective) tona­grams of 30 ragas (again cha­lana-s) to com­pute a clas­si­fi­ca­tion based on their tonal mate­r­i­al. Dissimilarities between pairs of graphs (cal­cu­lat­ed with Kuiper’s algo­rithm) were approx­i­mat­ed as dis­tances, from which a 3-dimensional clas­si­cal scal­ing was extracted:

A map of 30 North-Indian ragas con­struct­ed by com­par­ing tona­grams of 2-minute sketch­es (cha­lana-s) of sung per­for­mances (Bel 1988b)

This exper­i­ment sug­gests that con­tem­po­rary North-Indian ragas are amenable to mean­ing­ful auto­mat­ic clas­si­fi­ca­tion on the sole basis of their (time-independent) inter­val­ic con­tent. This approach is anal­o­gous to tech­niques of human face recog­ni­tion able to iden­ti­fy relat­ed images with the aid of lim­it­ed sets of features.

Microtonal framework

The “flex­i­ble” mod­el derived from the the­o­ret­i­cal mod­el of Natya Shastra (read The Two-vina exper­i­ment) dis­cards the asser­tion of a pre­cise fre­quen­cy ratio for the har­mon­ic major third clas­si­fied as anu­va­di (aso­nant) in ancient lit­er­a­ture. This amounts to admit­ting that the syn­ton­ic com­ma (pramāņa ṣru­ti in Sanskrit) might take any val­ue between 0 and 56.8 cents.

Let us look at graph­ic rep­re­sen­ta­tions (by the Bol Processor) to illus­trate these points.

The basic frame­work of musi­cal scales, accord­ing to Indian musi­col­o­gy, is a set of 22 tonal posi­tions in the octave named shru­ti-s in ancient texts. Below is the frame­work dis­played by Bol Processor (micro­ton­al scale “gra­ma”) with a 81/80 syn­ton­ic com­ma. The names of posi­tions “r1_”, “r2_” etc fol­low the con­straints of low­er­case ini­tials and append­ing a under­line char­ac­ter to dis­tin­guish octave num­bers. Positions “r1” and “r2” are two options for locat­ing komal Re (“Db” or “re bemol”) where­as “r3” and “r4” des­ig­nate shud­dha Re (“D” or “re”) etc.

The “gra­ma” scale dis­play­ing 22 shruti-s accord­ing to the mod­el of Natya Shastra

The 22 shru­ti-s can be lis­tened to on page Just into­na­tion, a gen­er­al frame­work, keep­ing in mind (read above) that this is a frame­work and not a scale. No musi­cian would ever attempt to play or sing these posi­tions as “notes”!

What hap­pens if the val­ue of the syn­ton­ic com­ma is mod­i­fied? Below is the same frame­work with a com­ma of 0 cent. In this case, any “har­mon­ic posi­tion” — one whose frac­tion con­tained a mul­ti­ple of 5 — slides to its near­est Pythagorean neigh­bour (only mul­ti­ples of 3 and 2). The result is a “Pythagorean tun­ing”. On top of the cir­cle the remain­ing gap is a Pythagorean com­ma. Positions are slight­ly blurred because of mis­match­es linked with a very small inter­val (the schis­ma).

The “gra­ma scale” of 22 shruti-s with a syn­ton­ic com­ma of 0 cent.

The fol­low­ing is the frame­work with a syn­ton­ic com­ma of 56.8 cents (its upper limit):

The “gra­ma scale” of 22 shruti-s with a syn­ton­ic com­ma of 56.8 cents.

In this rep­re­sen­ta­tion, “har­mon­ic major thirds” of 351 cents would most like­ly sound “out of tune” because the 5/4 ratio yields 384 cents. In fact, “g2” and “g3” are both dis­tant by a quar­ter­tone between Pythagorean “g1” (32/27) and Pythagorean “g4” (81/64). Nonetheless, the inter­nal con­sis­ten­cy of this frame­work (count­ing per­fect fifths in blue) makes it still eli­gi­ble for the con­struc­tion of musi­cal scales.

Between these lim­its of 0 and 56.8 cents, the graph­ic rep­re­sen­ta­tion of scales and their inter­nal tonal struc­ture remain unchanged if we keep in mind that the size of major-third inter­vals is decid­ed by the syn­ton­ic comma.

Construction of scale types

Manuscript of the descrip­tion of Zarlino’s “nat­ur­al” scale

The mod­el extract­ed from Bharata’s Natya Shastra is not an evi­dent ref­er­ence for pre­scrib­ing raga into­na­tion because this musi­cal genre start­ed its exis­tence a few cen­turies later.

Most of the back­ground knowl­edge required for the fol­low­ing pre­sen­ta­tion is bor­rowed from Bose (1960) and my late col­league E. James Arnold who pub­lished A Mathematical mod­el of the Shruti-Swara-Grama-Murcchana-Jati System (Journal of the Sangit Natak Akademi, New Delhi 1982). Arnold stud­ied Indian music in Banaras and Delhi dur­ing the 1970s and the ear­ly 1980s.

Bose was con­vinced (1960 p. 211) that the scale named Kaishika Madhyama is equiv­a­lent to a “just-intonation” seven-grade scale of Western musi­col­o­gy. In oth­er words, he took for grant­ed that the 5/4 fre­quen­cy ratio (har­mon­ic major third) should be equiv­a­lent to the 7-shru­ti inter­val, but this state­ment had no influ­ence on the rest of his analysis.

Arnold (right) and Bel (left) demon­strat­ing shruti-s at the inter­na­tion­al East-West music con­fer­ence, Bombay 1983

Arnold (1982 p. 17) imme­di­ate­ly used inte­ger ratios to design inter­vals with the fixed syn­ton­ic com­ma (81/80), but as sug­gest­ed above this has no impact on his mod­el with respect to its struc­tur­al descrip­tion. He insist­ed on set­ting up a “geo­met­ri­cal mod­el” rather than a spec­u­la­tive descrip­tion based on num­bers as many authors (e.g. Alain Daniélou) had attempt­ed it. The most inno­v­a­tive aspect of Arnold’s study has been the use a cir­cu­lar slid­ing mod­el to illus­trate the match­ings of inter­vals in trans­po­si­tion process­es (mur­ccha­na-s) — see The Two-vina exper­i­ment.

Indeed it would be more con­ve­nient to keep express­ing all inter­vals in num­bers of shru­ti-s in com­pli­ance with the ancient Indian the­o­ry, but a machine needs met­ri­cal data to draw graph­ics of scales. For this rea­son we show graphs using a 81/80 syn­ton­ic com­ma, keep­ing in mind the option of mod­i­fy­ing this val­ue at a lat­er stage.

Sa-grama and Ma-grama accord­ing to Natya Shastra. Red and green seg­ments indi­cate perfect-fifth con­so­nance. Underlined note names indi­cate ‘flat’ positions.

The 22-shru­ti frame­work offers the pos­si­bil­i­ty of con­struct­ing 211 = 2048 chro­mat­ic scales, among which only 12 are “opti­mal­ly con­so­nant”, i.e. con­tain­ing only one wolf major fifth (small­er by 1 syn­ton­ic com­ma = 22 cents).

The build­ing blocks of the tonal sys­tem accord­ing to tra­di­tion­al Indian musi­col­o­gy are two seven-grade scales named Ma-grama and Sa-grama. Bose wrote (1960 p. 13): the Shadja Grāma devel­oped from the ancient tetra­chord in which the hymns of the Sāma Veda were chant­ed. Later on anoth­er scale, called the Madhyama Grāma, was added to the sec­u­lar musi­cal sys­tem. The two scales (Dorian modes, accord­ing to Western ter­mi­nol­o­gy) dif­fer by the posi­tion of Pa (“G” or “sol”) which may dif­fer by a syn­ton­ic com­ma (pramāņa ṣru­ti). In the Sa-grama, inter­val Sa-Pa is a per­fect fifth (13 shru­ti-s) where­as it is a wolf fifth (12 shru­ti-s) in the Ma-grama. Conversely, inter­val Pa-Re is a per­fect fifth in Ma-grama and a wolf fifth in Sa-grama.

Bharata used the Sa-grama to expose his thought exper­i­ment (The Two vinas) aimed at deter­min­ing the sizes of shru­ti-s. Then he intro­duced two addi­tion­al notes: kakali Nishada (komal Ni or “Bflat”) and antara Gandhara (shud­dh Ga or “E”) to get a nine-grade scale from which “opti­mal­ly con­so­nant” chro­mat­ic scales could be derived from modal trans­po­si­tions (mur­ccha­na). The process of build­ing these 12 chro­mat­ic scales, name­ly “Ma01″, “Ma02″… “Sa01″, “Sa20″ etc. is explained on page Just into­na­tion, a gen­er­al frame­work.

Selecting notes in each chro­mat­ic scale yields 5 to 7-note melod­ic types. In the Natya Shastra these melod­ic types were named jāti. These may be seen as ances­tors of ragas even though their lin­eages and struc­tures are only spec­u­lat­ed (read on). The term thāṭ (pro­nounce ‘taat’) trans­lat­ed as “mode” or “par­ent scale” — has lat­er been adopt­ed, each thāṭ being called by the name of a raga (see Wikipedia). Details of the process, ter­mi­nol­o­gy and sur­veys of sub­se­quent musi­co­log­i­cal lit­er­a­ture will be found in pub­li­ca­tions by Bose and oth­er scholars.

The con­struc­tion of the basic scale types is explained by Arnold (1982 p. 37-38). The start­ing point is the chro­mat­ic Ma-grama in its basic posi­tion — name­ly “Sa_murcchana” in the “-cs.12_scales” Csound resource file. This scale can be visu­al­ized, using Arnold’s slid­ing mod­el, by plac­ing the S note of the inner wheel on the S of the out­er crown :

The Ma-grama chro­mat­ic scale in its basic posi­tion named “Sa_murcchana’

This yields the fol­low­ing intervals:

The Ma-grama chro­mat­ic scale in its basic posi­tion and with notes labeled in English

“Optimal con­so­nance” is illus­trat­ed by two fea­tures: 1) there is only one wolf fifth (red line) in the scale (between D and G), and 2) every note is con­nect­ed with anoth­er one by a per­fect fifth (blue line). This con­so­nance is of pri­or impor­tance to Indian musi­cians. Consonant inter­vals are casu­al­ly placed in melod­ic phras­es to enhance the “fla­vor” of their notes, and no wolf fifth should exist in the scale.

Note that the Ma-grama chro­mat­ic scale has all its notes in their low­er enhar­mon­ic position.

The Ma-grama chro­mat­ic scale has been renamed “Sa_murcchana” in this occur­rence because ‘S’ of the mov­ing wheel is fac­ing ‘S’ of the fixed crown. The names of notes have been (in a sin­gle click) con­vert­ed to the Indian con­ven­tion. Note that key num­bers also have been (auto­mat­i­cal­ly) fixed to match exclu­sive­ly labeled notes. In this way, the upper “sa” is assigned key 72 instead of 83 in the “Ma01″ scale showed on page Just into­na­tion, a gen­er­al frame­work. The tonal con­tent of this “Sa_murchana” is exposed on this table:

Tonal con­tent of “Sa_murcchana”
Scale type named “kaphi1”

Selecting only “unal­tered” notes in “Sa_murcchana” — sa, re, gak, ma, pa, dha, nik — yields the “kaphi1″ scale type named after raga Kaphi (pro­nounced ‘kafi’). This may be asso­ci­at­ed to a D-mode (Dorian) in Western musicology.

This scale type is stored under the name “kaphi1″ because there will be one more ver­sion of the Kaphi scale type.

In “Sa_murcchana” the selec­tion of notes can dif­fer in two ways:

  • Select antara Gandhara (name­ly “ga”) in replace­ment of the scale’s Gandhara (name­ly “gak”), there­by rais­ing it by 2 shru­ti-s. This yields a vikrit (mod­i­fied) scale type, name­ly “khamaj1″ asso­ci­at­ed with raga Khamaj.
  • Select both antara Gandhara and kakali Nishada (name­ly “ni” in replace­ment of “nik” raised by 2 shru­ti-s) which cre­ates the “bilaval1″ scale type asso­ci­at­ed with raga Bilaval.
A scale type named “bilaval3” match­ing Zarlino’s “nat­ur­al” scale

This “bilaval1″ scale type is one among three ver­sions of Bilaval cre­at­ed by the mur­ccha­na pro­ce­dure. Although it match­es the scale of white keys on a Western key­board instru­ment, it is not the com­mon “just into­na­tion” dia­ton­ic scale because of a wolf fifth between “sa” and “pa”.

An alter­nate Bilaval scale type named “bilaval3″ (extract­ed from “Ni1_murcchana”, see below) does match Giozeffo Zarlino’s “nat­ur­al” scale — read Just into­na­tion: a gen­er­al frame­work. This should not be con­fused with Zarlino’s mean­tone tem­pera­ment dis­cussed on page Microtonality.

An incom­plete­ly con­so­nant scale type

A fourth option: rais­ing “nik” to “ni” and keep­ing “gak”, would pro­duce a scale type in which “ni” does not have any con­so­nant rela­tion with anoth­er note of the scale. This option is there­fore dis­card­ed from the model.

Every mur­ccha­na of the Ma-grama chro­mat­ic scale pro­duces at least three scale types by select­ing unal­tered notes, antara Gandhara or both antara Gandhara and kakali Nishada.

Practically, to cre­ate for instance “Ni1_murcchana”, open the “Sa_murcchana” page and enter “nik” (i.e. N3) as the note to be placed on “sa”.

Raga scale types are stored in the “-cs.raga” Csound resource file. Images are avail­able in a sin­gle click and scale struc­tures are com­pared on the main page.

The entire process is sum­ma­rized in the fol­low­ing table (Arnold 1982 p. 38):

StepMa-grama chro­mat­ic
mur­ccha­na start­ing from
Shuddha gra­maVikrit gra­ma (antara)Vikrit gra­ma
(antara + kakali)
1Sakaphi1khamaj1bilaval1
2Ma1khamaj2bilaval2kalyan1
3Ni1bilaval3kalyan2marva1
4Ga1kalyan3marva2purvi1
5Dha1marva3purvi2todi1
6Re1purvi3todi2
7Ma3todi3lalit1
bhairao1
8Ni3lalit2
bhairao2
bhairavi1
9Ga3todi4
bhairavi2
10Dha3bhairavi3asavari1
11Re3bhairavi4asavari2kaphi2
12Pa3asavari3kaphi3khamaj3
Scale types of the extend­ed grama-murcchana series (Arnold 1982)

Usage of this table deserves a graph­ic demon­stra­tion. Let us for instance cre­ate scale type “kalyan1″ based on the “Ma1_murcchana”. The table says that both “antara and kakali” should be select­ed. This means “antara Gandhara” which is “ga” in replace­ment of “gak” in the Ma-grama scale, and “kakali Nishada” which is “ni” in replace­ment of “nik” in the Ma-grama scale. This process is clear on the mov­able wheel model:

Selecting notes to cre­ate the “kalyan1” scale type from the “Ma1_murcchana” of chro­mat­ic Ma-grama. “M1” is placed on “S”. Then the stan­dard inter­vals are picked up from the Ma-grama mov­ing wheel, replac­ing G1 with G3 and N1 with N3 as indi­cat­ed in the table.

To exe­cute this selec­tion and export the “kalyan1″ scale type, fill the form on page “Ma1_murcchana” as indi­cat­ed on the picture.

Below is the result­ing scale type.

The “kalyan1” scale type

Keep in mind that note posi­tions expressed as inte­ger fre­quen­cy ratios are just a mat­ter of con­ve­nience for read­ers acquaint­ed with Western musi­col­o­gy. It would be more appro­pri­ate to fol­low the Indian con­ven­tion of count­ing inter­vals in num­bers of shru­ti-s. In this exam­ple, the inter­val between “sa” and “ma” raised from 9 shru­ti-s (per­fect fourth) to 11 shru­ti-s (tri­tone).

Arnold’s mod­el is an exten­sion of the mur­ccha­na sys­tem described in Natya Shastra because it accepts mur­ccha­na-s start­ing from notes which do not belong to the orig­i­nal (7-grade) Ma-grama, tak­en from its “chro­mat­ic ver­sion”: Dha1, Re1, Ma3, Ni3, Ga3. This exten­sion is nec­es­sary for cre­at­ing scale types for Todi, Lalit and Bhairao which include aug­ment­ed sec­onds.

In his 1982 paper (p. 39-41) Arnold con­nect­ed his clas­si­fi­ca­tion of scale types with the tra­di­tion­al list of jāti-s, the “ances­tors of ragas” described in Sangita Ratnakara of Śārṅgadeva (Shringy & Sharma, 1978). Seven jāti-s are cit­ed (p. 41), each of them being derived from a mur­ccha­na of Ma-grama on one of its shud­dha swara-s (basic notes). 

Every jāti is assigned a note of ten­sion release (nyasa swara). In con­tem­po­rary ragas, the nyasa swara is often found at the end of a phrase or a set of phras­es. In Arnold’s inter­pre­ta­tion, the same should define the mur­ccha­na from which the melod­ic type (jāti) is born. Since, in fact, the names of the shud­dha jatis are tied to their nyasa swaras, this too sug­gests that they should be tied to the mur­ccha­nas belong­ing from those nyasa swaras (Arnold 1982 p. 40).

Performance times asso­ci­at­ed with murcchana-s of the Ma-grama, accord­ing to Arnold (1985)

In oth­er pub­li­ca­tions (notably Arnold & Bel 1985), Arnold used the cycle of 12 chro­mat­ic scales to sug­gest that enhar­mon­ic posi­tions of the notes might express ten­sions or release states bound to the chang­ing ambi­ence of the cir­ca­di­an cycle, there­by pro­vid­ing an expla­na­tion of per­for­mance times assigned to tra­di­tion­al ragas. Low enhar­mon­ic posi­tions would be asso­ci­at­ed with dark­ness and high­er ones with day light. In this way, ragas con­struct­ed with the aid of the Sa mur­ccha­na of Ma-grama chro­mat­ic scale (all low posi­tions, step 1) might be inter­pret­ed near mid­night where­as the ones mix­ing low and high posi­tions (step 7) would car­ry the ten­sions of sun­rise and sun­set. Their suc­ces­sion is a cycle because, in the table shown above, it is pos­si­ble to jump from step 12 to step 1 by low­er­ing all note posi­tions by one shru­ti. This cir­cu­lar­i­ty is implied by the process named sadja-sadharana in musi­co­log­i­cal lit­er­a­ture (Shringy & Sharma 1978).

A list of 85 ragas with per­for­mance times pre­dict­ed by the mod­el is avail­able in Arnold & Bel (1985). This hypoth­e­sis is indeed inter­est­ing — and it does hold for many well-known ragas — but we could nev­er embark on a sur­vey of musi­cians’ state­ments about per­for­mance times that might have assessed its validity.

Practice

Given scale types stored in the “-cs.raga” Csound resource file, Bol Processor + Csound can be used to check the valid­i­ty of scales by play­ing melodies of ragas they are sup­posed to embody. It is also inter­est­ing to use these scales in musi­cal gen­res unre­lat­ed with North Indian raga and dis­tort them in any imag­in­able direction…

Choice of a raga

Todi Ragini, Ragamala, Bundi, Rajasthan, 1591
Public domain

We will take the chal­lenge of match­ing one among the four “todi” scales with two real per­for­mances of raga Todi.

Miyan ki todi is present­ly the most impor­tant raga of the Todi fam­i­ly and there­fore often sim­ply referred to as Todi […], or some­times Shuddh Todi. Like Miyan ki mal­har it is sup­posed to be a cre­ation of Miyan Tansen (d. 1589). This is very unlike­ly, how­ev­er, since the scale of Todi at the time of Tansen was that of mod­ern Bhairavi (S R G M P D N), and the name Miyan ki todi first appears in 19th cen­tu­ry lit­er­a­ture on music.

Joep Bor (1999)

This choice is a chal­lenge for sev­er­al rea­sons. Among them, the four vari­ants of “todi” scales have been dri­ven from a (ques­tion­able) exten­sion of the grama-murcchana sys­tem. Then, notes “ni” and “rek”, “ma#” and “dhak” are close to the ton­ic “sa” and the dom­i­nant “pa” and might be “attract­ed” by the ton­ic and dom­i­nant, there­by dis­rupt­ing the “geom­e­try” of the­o­ret­i­cal scales in the pres­ence of a drone.

Finally, and most impor­tant, the per­former’s style and per­son­al options are expect­ed to come in con­tra­dic­tion with this the­o­ret­i­cal mod­el. As sug­gest­ed by Rao and van der Meer (2010 p. 693):

[…] it has been observed that musi­cians have their own views on into­na­tion, which are hand­ed down with­in the tra­di­tion. Most of them are not con­scious­ly aware of aca­d­e­m­ic tra­di­tions and hence are not in a posi­tion to express their ideas in terms of the­o­ret­i­cal for­mu­la­tions. However, their ideas are implic­it in musi­cal prac­tice as musi­cians visu­al­ize tones, per­haps not as fixed points to be ren­dered accu­rate­ly every time, but rather as tonal regions or pitch move­ments defined by the gram­mar of a spe­cif­ic raga and its melod­ic con­text. They also attach para­mount impor­tance to cer­tain raga-specific notes with­in phras­es to be intoned in a char­ac­ter­is­tic way.

We had already tak­en the Todi chal­lenge with an analy­sis of eight occur­rences using the Melodic Movement Analyzer (Bel 1988). The ana­lyz­er had pro­duced streams of accu­rate pitch mea­sure­ments which were sub­mit­ted to a sta­tis­ti­cal analy­sis after being fil­tered as selec­tive tona­grams (Bel 1984; Bel & Bor 1984). Occurrences includ­ed 6 per­for­mances of raga Todi and 2 exper­i­ments of tun­ing the Shruti Harmonium.

The MMA analy­sis revealed a rel­a­tive­ly high con­sis­ten­cy of note posi­tions show­ing stan­dard devi­a­tions bet­ter than 6 cents for all notes except “ma#” for which the devi­a­tion rose to 10 cents, still an excel­lent sta­bil­i­ty. Matching these results against the grama-murcchana “flex­i­ble” mod­el revealed less than 4 cent stan­dard devi­a­tion of inter­vals for 4 dif­fer­ent scales in which the syn­ton­ic com­ma would be adjust­ed to 6, 18, 5 and 5 cents. In dis­cussing tun­ing schemes we even envis­aged that musi­cians might “solve the prob­lem” of a “ni-ma#” wolf fifth by tem­per­ing fifths over the “ni-ma#-rek-dhak” cycle.

Our con­clu­sion was that no par­tic­u­lar “tun­ing scheme” could be tak­en for grant­ed on the basis of “raw” data. It would be more real­is­tic to study a par­tic­u­lar per­for­mance by a par­tic­u­lar musician.

Choice of a musician

Kishori Amonkar per­form­ing raga Lalit. Credit সায়ন্তন ভট্টাচার্য্য - Own work, CC BY-SA 4.0

Working with the Shruti Harmonium nat­u­ral­ly incit­ed us to meet Kishori Amonkar in 1981. She was a fore­most expo­nent of Hindustani music, hav­ing devel­oped a per­son­al style that claimed to tran­scend clas­si­cal schools (gha­ranas).

Most inter­est­ing, she used to per­form with the accom­pa­ni­ment of a swara man­dal (see pic­ture), a zither which she would tune for each indi­vid­ual raga. We were not equipped for mea­sur­ing these tun­ings with a suf­fi­cient accu­ra­cy. Therefore we brought the Shruti Harmonium to pro­gram inter­vals as per her instructions.

This did not work well for two rea­sons. A tech­ni­cal one: that day, a fre­quen­cy divider LSI cir­cuit was defec­tive on the har­mo­ni­um; until it was replaced some pro­grammed inter­vals were inac­ces­si­ble. A musi­cal one: the exper­i­ment revealed that this accu­rate har­mo­ni­um was unfit to tun­ing exper­i­ments with Indian musi­cians. Frequency ratios need­ed to be typed on a small key­board, a usage too remote from the con­text of per­for­mance. This was a major incen­tive for design­ing and con­struct­ing a “micro­scope for Indian music”, the Melodic Movement Analyzer (MMA) (Bel & Bor 1984).

During the fol­low­ing years (1981-1984) MMA exper­i­ments took our entire time, reveal­ing the vari­abil­i­ty (yet not the ran­dom­ness) of raga into­na­tion. For this rea­son we could not return to tun­ing exper­i­ments. Today, a sim­i­lar approach would be much eas­i­er with the help of Bol Processor BP3… if only the expert musi­cians of that peri­od were still alive!

Choice of a scale type

We need to decide between the four “todi” scale types pro­duced by mur­ccha­na-s of the Ma-grama chro­mat­ic scale. To this effect we may use mea­sure­ments by the Melodic Movement Analyzer (Bel 1988 p. 15). Let us pick up aver­age mea­sure­ments and the ones of a per­for­mance of Kishori Amonkar. These are note posi­tions (in cents) against the ton­ic “sa”.

NoteAverageStandard devi­a­tionKishori Amonkar
rek95496
gak2944288
ma#60610594
pa7021702
dhak7923792
(dhak)8063810
ni110761110
The “dhak” between brack­ets is a mea­sure­ment on the low octave

For the moment we ignore “dhak” in the low­er octave as it will be dealt with sep­a­rate­ly. Let us match Kishori Amonkar’s results with the four scale types:

NoteKishori Amonkartodi1todi2todi3todi4
rek96898989112
gak288294294294294
ma#594590590610610
pa702702702700702
dhak792792792792814
ni11101088110911091109
Scale type “todi2”, the best match to a per­for­mance of Kishori Amonkar

There are sev­er­al ways of find­ing the best match for musi­cal scales: either com­par­ing scale inter­vals or com­par­ing note posi­tions with respect to the base note. Due to the impor­tance of the drone we opt for the sec­ond method. The selec­tion is easy here. Version “todi1″ may be dis­card­ed because of “ni”, the same with “todi3″ and “todi4″ because of “ma#”. We are left with “todi2″ which has a very good match­ing, includ­ing with the mea­sure­ments of per­for­mances by oth­er musicians.

Adjustment of the scale

The largest devi­a­tions are on “rek” which was per­formed 7 cents high­er than the pre­dict­ed val­ue and “gak” 6 cents low­er. Even a 10-cent vari­a­tion is prac­ti­cal­ly impos­si­ble to mea­sure on a sin­gle note sung by a human, includ­ing a high-profile singer like Kishori Amonkar; the best res­o­lu­tion used in speech prosody is larg­er than 12 cents.

Any “mea­sure­ment” of the MMA is an aver­age of val­ues along the rare sta­ble melod­ic steps. It may not be rep­re­sen­ta­tive of the “real” note because of its depen­den­cy on note treat­ment: if the approach of the note lies in a range on the lower/higher side, the aver­age will be lower/higher than the tar­get pitch.

Therefore it would be accept­able to declare that the “todi2″ scale type match­es the per­for­mance. Nonetheless, let us demon­strate ways of mod­i­fy­ing the mod­el to reflect the mea­sure­ments more accurately.

First we dupli­cate “todi2″ to cre­ate “todi-ka” (see pic­ture). Note posi­tions are iden­ti­cal in both versions.

Looking at the pic­ture of the scale (or fig­ures on its table) we notice that all note posi­tions except “ma#” are Pythagorean. The series which a note belongs to is marked by the col­or of its point­er: blue for Pythagorean and green for harmonic.

Modified “todi2” scale match­ing the mea­sured “ma#”

This means that mod­i­fy­ing the size of the syn­ton­ic com­ma — in strict com­pli­ance with the grama-murcchana mod­el — will only adjust “ma#”. In order to change “ma#” posi­tion from 590 to 594 cents (admit­ted­ly a ridicule adjust­ment) we need to decrease the size of the syn­ton­ic com­ma by the same amount. This can be done at the bot­tom right of the “todi-ka” page, chang­ing the syn­ton­ic com­ma to 17.5 cents, a mod­i­fi­ca­tion which is con­firmed by the new picture.

A table on the “todi-ka” page indi­cates that the “rek-ma#” inter­val is still a per­fect fifth even though it is small­er by 6 cents.

It may not be evi­dent whether the syn­ton­ic com­ma needs to be increased or decreased to fix the posi­tion of “ma#”, but it is easy to try the oth­er way in case the direc­tion was wrong. 

Final ver­sion of “todi2” adjust­ed to Kishori Amonkar’s per­for­mance in the medi­um octave (4)

Other adjust­ments will depart from the “pure” mod­el. These lead to chang­ing fre­quen­cy ratios in the table of the “todi-ka” page. Raising “rek” from 89 to 96 cents requires a rais­ing of 7 cents amount­ing to ratio 2(7/1200) = 1.00405. This brings the posi­tion of “rek” from 1.053 to 1.057.

In the same way, low­er­ing “gak” from 294 to 288 cents requires a low­er­ing of 6 cents amount­ing to ratio 2(-6/1200) = 0.9965. This brings the posi­tion of “gak” from 1.185 to 1.181.

Fortunately, these cal­cu­la­tions are done by the machine: use the “MODIFY NOTE” but­ton on the scale page.

The pic­ture shows that the infor­ma­tion of “rek” and “gak” belong­ing to Pythagorean series (blue line) is pre­served. The rea­son is that when­ev­er a fre­quen­cy ratio is mod­i­fied by its floating-point val­ue, the machine ver­i­fies whether the new val­ue comes close to an inte­ger ratio of the same series. For instance, chang­ing back “rek” to 1.053 would restore its ratio 256/243. Accuracy bet­ter than 1‰ is required for this matching.

A tun­ing scheme for this scale type is sug­gest­ed by the machine. The graph­ic rep­re­sen­ta­tion shows that “ni” is not con­so­nant with “ma#” as their inter­val is 684 cents, close to a wolf fifth of 680 cents. Other notes are arranged on two cycles of per­fect fifths. Interestingly, rais­ing “rek” by 7 cents brought the “rek-ma#” fifth back to its per­fect size (702 cents).

Again, these are mean­ing­less adjust­ments for a vocal per­for­mance. We are only show­ing how to pro­ceed when necessary.

The “todi2” scale type with “dhak” adjust­ed for the low octave (3)

The remain­ing adjust­ment will be that of “dhak” in the low­er octave. To this effect we dupli­cate the pre­ced­ing scale after renam­ing it “todi_ka_4″, indi­cat­ing that it is designed for the 4th octave. In the new scale named “todi_ka_3″, we raise “dhak3” by 810 -792 = 18 cents.

This rais­es its posi­tion from 1.58 to 1.597. Note that this brings it exact­ly to a posi­tion in the har­mon­ic series since the syn­ton­ic com­ma is 17.5 cents.

In addi­tion, “dhak-sa” is now a har­mon­ic major third — with a size of 390 cents fit­ting the 17.5 cents com­ma. This is cer­tain­ly mean­ing­ful in the melod­ic con­text of this raga, a rea­son why an adjust­ment of the same size had been done by all musi­cians in their per­for­mances or tun­ing experiments.

This case is a sim­ple illus­tra­tion of raga into­na­tion as a trade-off between har­monic­i­ty with respect to the drone and the require­ment of con­so­nant melod­ic inter­vals. It also indi­cates that the Shruti Harmonium could not fol­low musi­cians’ prac­tice because its scale ratios were repli­cat­ed in all octaves.

Choice of a recording

We don’t have the record­ing on which the MMA analy­sis had been done. A prob­lem with old tape record­ings is the unre­li­a­bil­i­ty of speed in tape trans­porta­tion. On a long record­ing, too, the fre­quen­cy of the ton­ic may change a lit­tle due to vari­a­tions of room tem­per­a­ture influ­enc­ing instru­ments — includ­ing tape dilation…

To try match­ing scales a with real per­for­mances and exam­ine extreme­ly small “devi­a­tions” (which have lit­tle musi­cal sig­nif­i­cance, in any) it is there­fore safer to work with dig­i­tal record­ings. This was the case with Kishori Amonkar’s Todi record­ed in London in the ear­ly 2000 for the Passage to India col­lec­tion and avail­able free of copy­right (link on Youtube). The fol­low­ing is based on that recording.

Setting up the diapason

Let us cre­ate the fol­low­ing “-gr.tryRagas” gram­mar:

-se.tryRagas
-cs.raga

S --> _scale(todi_ka_4,0) sa4

Adjusting note con­ven­tion in “-se.tryRagas”

In “-se.tryRagas” the note con­ven­tion should be set to “Indian” so that “sa4” etc. is accept­ed even when no scale is specified.

The gram­mar calls “-cs.raga” con­tain­ing the def­i­n­i­tions of all scale types cre­at­ed by the pro­ce­dure described above. Unsurprisingly, it does not play note “sa” at the fre­quen­cy of the record­ing. We there­fore need to mea­sure the ton­ic to adjust the fre­quen­cy of “A4” (dia­pa­son) in “-se.tryRagas” accord­ing­ly. There are sev­er­al ways to achieve this with increas­ing accuracy.

A semi­tone approx­i­ma­tion may be achieved by com­par­ing the record­ing with notes played on a piano or any elec­tron­ic instru­ment tuned with A4 = 440 Hz. Once we have found the key that is clos­est to “sa” we cal­cu­late its fre­quen­cy ratio to A4. If the key is F#4, which is 3 semi­tones low­er than A4, the ratio is r = 2(-3/12) = 0.840. To get this fre­quen­cy on “sa4” we there­fore would need to adjust the fre­quen­cy of the dia­pa­son (in “-se.tryRagas”) to:

440 * r * 2(9/12) = 440 * 2((9-3)/12) = 311 Hz

A much bet­ter approx­i­ma­tion is achieved by extract­ing a short occur­rence of “sa4” at the very begin­ning of the performance:

A short occur­rence of “sa4” in the begin­ning of Kishori Amonkar’s raga Todi

Then select a seem­ing­ly sta­ble seg­ment and expand the time scale to get a vis­i­ble signal:

Expansion of a very brief “sta­ble” occur­rence of “sa4”

This sam­ple con­tains 9 cycles for a dura­tion of 38.5 ms. The fun­da­men­tal fre­quen­cy is there­fore 9 * 1000 / 38.5 = 233.7 Hz. Consequently, adjust the dia­pa­son in “-se.tryRagas” to 233.7 * 2(9/12) = 393 Hz.

The last step is a fine tun­ing com­par­ing by ear the pro­duc­tion of notes in the gram­mar with the record­ing of “sa4” played in a loop. To this effect we pro­duce the fol­low­ing sequence:

S --> _pitchrange(500) _tempo(0.2) Scale _pitchbend(-15) sa4 _pitchbend(-10) sa4 _pitchbend(-5) sa4 _pitchbend(-0) sa4 _pitchbend(+5) sa4 _pitchbend(+10) sa4 _pitchbend(+15) sa4 _pitchbend(+20) sa4

These are eight occur­rences of “sa4” played at slight­ly increas­ing pitch­es adjust­ed by the pitch­bend. First make sure that the pitch­bend is mea­sured in cents: this is indi­cat­ed in instru­ment “Vina” called by “-cs.raga” and Csound orches­tra file “new-vina.orc”.

Listening to the sequence may not reveal pitch dif­fer­ences, but these will appear to a trained ear when super­posed with the recording:

Recording on “sa4” super­posed with a sequence of “sa4” at slight­ly increas­ing pitch­es. Which occur­rence is in tune?
➡ This is a stereo record­ing. Use ear­phones to hear the music and sequence of plucked notes separately

One of the four occur­rences sounds best in tune. Suppose that the best match is on _pitchbend(+10). This means that the dia­pa­son should be raised by 10 cents. Its new fre­quen­cy would there­fore be 393 * 2(10/1200) = 395.27 Hz.

In fact the best fre­quen­cy is 393.22 Hz, which amounts to say­ing that the sec­ond eval­u­a­tion (yield­ing 393 Hz) was fair — and the singers’ voic­es very reli­able! Now we can ver­i­fy the fre­quen­cy of “sa4” on the Csound score:

; Csound score
f1 0 256 10 1 ; This table may be changed
t 0.000 60.000
i1 0.000 5.000 233.814 90.000 90.000 0.000 -15.000 -15.000 0.000 ; sa4
i1 5.000 5.000 233.814 90.000 90.000 0.000 -10.000 -10.000 0.000 ; sa4
i1 10.000 5.000 233.814 90.000 90.000 0.000 -5.000 -5.000 0.000 ; sa4
i1 15.000 5.000 233.814 90.000 90.000 0.000 0.000 0.000 0.000 ; sa4
i1 20.000 5.000 233.814 90.000 90.000 0.000 5.000 5.000 0.000 ; sa4
i1 25.000 5.000 233.814 90.000 90.000 0.000 10.000 10.000 0.000 ; sa4
i1 30.000 5.000 233.814 90.000 90.000 0.000 15.000 15.000 0.000 ; sa4
i1 35.000 5.000 233.814 90.000 90.000 0.000 20.000 20.000 0.000 ; sa4
s

These meth­ods could in fact be sum­ma­rized by the third one: use the gram­mar to pro­duce a sequence of notes in a wide range to deter­mine an approx­i­mate pitch of “sa4” until the small range for the pitch­bend (± 200 cents) is reached. Then play sequences with pitch­bend val­ues in increas­ing accu­ra­cy until no dis­crim­i­na­tion is possible.

In a real exer­cise it would be safe to check the mea­sure­ment of “sa4” against occur­rences in sev­er­al parts of the recording.

This approach is indeed too demand­ing on accu­ra­cy for the analy­sis of a vocal per­for­mance, but it will be appre­cia­ble when work­ing with a long-stringed instru­ment such as the rudra veena. We will show it with Asad Ali Kan’s per­for­mance.

Matching phrases of the performance

We are now ready to check whether note sequences pro­duced by the mod­el would match sim­i­lar sequences of the recording.

We first try a sequence with empha­sis on “rek”. The fol­low­ing note sequence is pro­duced by the grammar:

S --> KishoriAmonkar1
KishoriAmonkar1 --> Scale _ {2, dhak3 sa4 ni3 sa4} {7, rek4} _ {2, dhak3 sa4 ni3 dhak3} {2, dhak3 _ ni3 sa4} {5, rek4}
Scale --> _scale(todi_ka_3,0)

Below is the phrase sung by the musi­cians (loca­tion 0′50″) then repeat­ed in super­po­si­tion with the sequence pro­duced by the grammar:

A phrase with empha­sis on “rek” sung by Kishori Amonkar, then repro­duced in super­po­si­tion with the sequence of notes pro­duced by the gram­mar using scale “todi_ka_3”
➡ This is a stereo record­ing. Use ear­phones to hear the music and sequence of plucked notes separately

In this exam­ple, scale “todi_ka_3″ has been used because of the occur­rence of brief instances of “dhak3”. The posi­tion of “rek” is iden­ti­cal in the 3d and 4th octaves. The blend­ing of voice with the plucked instru­ment is remark­able in the final held note.

In the next sequence (loca­tion 1′36″) the posi­tion of “gak4” will be appre­ci­at­ed. The gram­mar is the following:

S --> KishoriAmonkar2
KishoriAmonkar2 --> Scale {137/100, sa4 rek4 gak4 rek4} {31/10, rek4} {18/10, gak4} {75/100,rek4} {44/10, sa4}
Scale --> _scale(todi_ka_4,0)

A phrase tar­get­ing “gak” repeat­ed in super­po­si­tion with the sequence of notes pro­duced by the gram­mar using scale “todi_ka_4”

This time, the scale “todi_ka_4″ was select­ed, even though it had no inci­dence on the into­na­tion since “dhak” is absent.

A word about build­ing the gram­mar: we looked at the sig­nal of the record­ed phrase and mea­sured the (approx­i­mate) dura­tions of notes: 1.37s, 3.1s, 1.8s, 7.5s, 4.4s. Then we con­vert­ed these dura­tions to inte­ger ratios — frac­tions of the basic tem­po whose peri­od is exact­ly 1 sec­ond as per the set­ting in “-se.tryRagas”: 137/100, 31/10 etc.

Signal of the pre­ced­ing record­ed phrase

Below is a pianoroll of the sequence pro­duced by the grammar:

Pianoroll of the note sequence pro­duced by the grammar

No we try a phrase with a long rest on “dhak3” (loca­tion 3′34″) prov­ing that scale “todi_ka_3″ match­es per­fect­ly this occur­rence of “dhak”:

S --> KishoriAmonkar3
KishoriAmonkar3 --> scale(todi_ka_3,0) 11/10 {19/20, ma#3 pa3} {66/10,dhak3} {24/10, ni3 dhak3 pa3 }{27/10,dhak3} 12/10 {48/100,dhak3}{17/10,ni3}{49/10,dhak3}

A phrase rest­ing on “dhak3” repeat­ed in super­po­si­tion with the sequence of notes pro­duced by the gram­mar using scale “todi_ka_3”
Pianoroll of the note sequence pro­duced by the gram­mar with a rest on “dhak3”

Early occur­rence of “ma#4” (loca­tion 11′38″):

S --> KishoriAmonkar4
KishoriAmonkar4 --> _scale(todi_ka_4,0) 4/10 {17/10, ni3}{26/100,sa4}{75/100,rek4}{22/100,gak4}{17/10,ma#4}{16/100,gak4}{34/100,rek4}{56/100,sa4}{12/100,rek4}{84/100,gak4}{27/100,rek4}{12/10,sa4}

Early occur­rence of “ma#4”

Hitting “dhak4” (loca­tion 19′46″):

S --> KishoriAmonkar5
KishoriAmonkar5 --> _scale(todi_ka_4,0) 13/10 {16/10,ma#4}{13/10,gak4}{41/100,ma#4}{72/100,ma#4 dhak4 ma#4 gak4 ma#4}{18/10,dhak4}{63/100,sa4}{90/100,rek4}{30/100,gak4}{60/100,rek4}{25/100,sa4}{3/2,rek4}

Hitting “dhak4”…

With a light touch of “pa4” (loca­tion 23′11″):

S --> KishoriAmonkar6
KishoriAmonkar6 --> _scale(todi_ka_4,0) 28/100 {29/100,ma#4}{40/100,dhak4}{63/100,ni4 sa5 ni4}{122/100,dhak4}{64/100,pa4}{83/100,ma#4}{44/100,pa4}{79/100,dhak4}

A light touch of “pa”

Pitch accu­ra­cy is no sur­prise in per­for­mances by Kishori Amonkar. With a strong aware­ness of “shru­ti-s”, she would sit on the stage pluck­ing her swara man­dal care­ful­ly tuned for each raga.

A test with the rudra veena

Asad Ali Khan play­ing the rudra veena

Asad Ali Khan was one of the last per­form­ers of the rudra veena in the end of the 20th cen­tu­ry and a very sup­port­ive par­tic­i­pant in sci­en­tif­ic research on raga into­na­tion. Pitch accu­ra­cy is such on this instru­ment that we could iden­ti­fy tiny vari­a­tions con­trolled and sig­nif­i­cant in the con­text of the raga. Read for instance Playing with Intonation (Arnold 1985). In order to mea­sure vibra­tions below the range of audi­ble sounds, we occa­sion­al­ly fixed a mag­net­ic pick­up near the last string.

Below are the sta­tis­tics of mea­sure­ments by the Melodic Movement Analyzer of raga Miyan ki Todi inter­pret­ed by Asad Ali Khan in 1981. The sec­ond col­umn con­tains the mea­sure­ments of his tun­ing of the Shruti Harmonium dur­ing an exper­i­ment. Columns on the right dis­play pre­dict­ed note posi­tions accord­ing to the grama-murcchana mod­el with a syn­ton­ic com­ma of ratio 81/80. Again in this raga, “dhak” may take dif­fer­ent val­ues in the peer­for­mance, depend­ing on the octave.

NoteAsad Ali Khan
per­form­ing
Asad Ali Khan
tun­ing
todi1todi2todi3todi4
rek99100898989112
gak290294294294294294
ma#593606590590610610
pa702702702702700702
dhak3795794792792792814
dhak2802
ni110511081088110911091109

Again, the best match would be the “todi2″ scale with a syn­ton­ic com­ma of 17.5 cents. We cre­at­ed two scales, “todi_aak_2″ and “todi_aak_3″ for the 2nd and 3th octaves.

Adjustments of the “todi2” scale for Asad Ali Kan’s per­for­mance on the rudra veena. Low octave on the left and medi­um on the right.

The scale con­struct­ed dur­ing the Shruti Harmonium exper­i­ment is of less­er rel­e­vance because of the influ­ence of the exper­i­menter play­ing scale inter­vals with a low-attracting drone (pro­duced by the machine). In his attempt to resolve dis­so­nance in the scale — which always con­tained a wolf fifth and sev­er­al Pythagorean major thirds — Khan saheb end­ed up with a tun­ing iden­ti­cal to the ini­tial one but one com­ma low­er. This was not a musi­cal­ly sig­nif­i­cant situation!

Tuning scheme for “todi_aak_2”

Scale “todi_aak_2″ (in the low octave) con­tains inter­est­ing inter­vals (har­mon­ic major thirds) which lets us antic­i­pate effec­tive melod­ic move­ments. The tun­ing scheme sum­ma­rizes these relations.

We are now tak­ing frag­ments of Asad Ali Khan’s per­for­mance of Todi (2005) avail­able on Youtube (fol­low this link).

The per­for­mance began in the low octave, there­fore with scale “todi_aak_2″. The fre­quen­cy of Sa was mea­sured at 564.5 Hz with the method explained earlier.

Let us start with a sim­ple melod­ic phrase repeat­ed two times, the sec­ond time in super­po­si­tion with the note sequence pro­duced by the grammar.

A phrase of raga Todi by Asad Ali Khan repeat­ed 2 times, the sec­ond time in super­po­si­tion with the sequence of notes pro­duced by the gram­mar
➡ This is a stereo record­ing. Use ear­phones to hear the music and sequence of plucked notes separately

S --> AsadAliKhan1
AsadAliKhan1 --> _scale(todi_aak_2,0) 45/100 {69/10,sa3} {256/100,dhak2} {78/10,dhak2} {12/10,sa3 ni2 rek3&} {48/10,&rek3} {98/100,sa3 ni2 sa3&} {27/10,&sa3}

This gram­mar con­tains an unusu­al sign ‘&’ used to con­cate­nate sound-objects (or notes) beyond the bor­ders of poly­met­ric expres­sions (between curled brack­ets). This makes it pos­si­ble to play the final “rek3” and “sa3” as con­tin­u­ous notes. This con­ti­nu­ity is clear on the fol­low­ing graph:

The end of the phrase, show­ing “rek3” and “sa3” as con­tin­u­ous notes

It is time to make sure that accu­rate tun­ings and adjust­ments of scales are more than an intel­lec­tu­al exer­cise… After all, the main dif­fer­ence between scales “todi_aak_2″ and “todi_aak_3″ is that “dhak” is 7 cents high­er in “todi_aak_2″, which means a third of a com­ma! To check the effect of the fine tun­ing, lis­ten to the super­im­po­si­tion two times, once with “todi_aak_3″ and the sec­ond time with “todi_aak_2″:

The same “dhak2” with a note pro­duced using “todi_aak_3” and the sec­ond time “todi_aak_2”

To check the dif­fer­ence between these two ver­sions of “dhak2” we can play them in sequence, then superimposed:

S --> _tempo(1/2) _scale(todi_aak_3,0) dhak2 _scale(todi_aak_2,0) dhak2 {_scale(todi_aak_3,0) dhak2, _scale(todi_aak_2,0) dhak2}

The two ver­sions of “dhak2” in sequence then superimposed

With fun­da­men­tal fre­quen­cies 132.837 Hz and 133.341 Hz, the beat fre­quen­cy (of sine waves) would be 133.341 - 132.837 = 0.5 Hz. The per­ceived beat fre­quen­cy is high­er because of the inter­fer­ence between high­er par­tials. This sug­gests that a dif­fer­ence of 7 cents is not irrel­e­vant in the con­text of notes played by a long-stringed instru­ment (Arnold 1985).

More in the low­er octave:

S --> AsadAliKhan2
AsadAliKhan2 --> scale(todi_aak_2,0) _volume(64) _pitchrange(500) _pitchcont 93/100 {81/10,pa2}{38/10,pa2 gak2 pa2 dhak2 pa2 }{19/10,gak2}{43/10, _pitchbend(0) rek2 _pitchbend(-100) rek2&} _volumecont _volume(64) {2, _pitchbend(-100) &rek2} _volume(0) _volume(64) {23/10,ni2__ dhak2}{103/100,sa3&}{4,&sa3} 15/10 _volume(64) {38/10,sa3} _volume(0)

As “sa2” is out of range of the Csound instru­ment “Vina”, it is per­formed here as “rek2” with a pitch­bend cor­rec­tion of one semitone.

Low-octave phrase repeat­ed with attempt­ed super­im­po­si­tion of a note sequence

The ren­der­ing of phras­es in the low octave is very approx­i­ma­tive because of the pre­dom­i­nance of meend (pulling the string). Some effects could be bet­ter imi­tat­ed with the aid of per­for­mance con­trols — see for instance Sarasvati Vina — but this requires a mas­tery of the real instru­ment to design pat­terns of musi­cal “ges­tures” rather than sequences of sound events… Imitating the melod­ic intri­ca­cy of raga is not the top­ic of this page; we are mere­ly check­ing the rel­e­vance of scale mod­els to the “tonal skele­ton” of ragas.

Accidental notes

Raga scales extract­ed from mur­ccha­nas of the Ma-grama chro­mat­ic scale (see above) con­tain exclu­sive­ly notes pre­sum­ably belong­ing to the raga. They can­not accom­mo­date acci­den­tal notes nor the scales used by mix­ing ragas, a com­mon practice.

Let us take for instance a frag­ment of the pre­ced­ing exam­ple which was poor­ly ren­dered by the sequence of notes pro­duced by the gram­mar. (We learn from our mis­takes!) We may feel like replac­ing expres­sion {38/10, pa2 gak2 pa2 dhak2 _ pa2 _} with {38/10, pa2 ga2 pa2 dhak2 _ pa2 _} mak­ing use of “ga2” which does not belong to the “todi_aak_2″ scale. Unfortunately, this pro­duces an error message:

ERROR Pitch class ‘4’ does not exist in _scale(todi_aak_2). No Csound score produced.

This amounts to say­ing that scale “todi2″ con­tains no map­ping of key #64 to “ga” — nor key # 65 to “ma”, see picture.

To solve this prob­lem we may recall that scale “todi2″ has been extract­ed from “Re1_murcchana”. The lat­ter con­tains all grades of a chro­mat­ic scale in addi­tion to the extract­ed ones. Therefore it is suf­fi­cient to replace “_scale(todi_aak_2,0)” with “_scale(Re1_murcchana,0)” in this section:

_scale(Re1_murcchana,0) {38/10, pa2 ga2 pa2 dhak2 _ pa2 _} _scale(todi_aak_2,0) etc.

The scale edi­tor takes care of assign­ing each note a key num­ber based on the chro­mat­ic scale if a stan­dard English, Italian/French or Indian note con­ven­tion is used. In oth­er cas­es this map­ping should be done by hand. Designers of micro­ton­al scales should stay aware of key map­pings if they use cus­tomized names for “notes”.

Another prob­lem aris­es because in “todi_aak_2″ note “dhak” had been raised from 792 to 810 cents, which is not its val­ue in “Re1_murcchana”. This may be fixed by cre­at­ing anoth­er vari­ant of the scale with this cor­rec­tion, or sim­ply use the pitch­bend to mod­i­fy “dhak2” — in which case the same pitch­bend could have been used in the first place to raise “gak2”.

Finally, the best approach to avoid this prob­lem would be to use the source chro­mat­ic scale “Re1_murcchana”, a mur­ccha­na of Ma-grama, to con­struct raga scales even though some grades will nev­er be used.

To conclude…

This whole dis­cus­sion was tech­ni­cal. There is no musi­cal rel­e­vance in try­ing to asso­ciate plucked notes with very sub­tly orna­ment­ed melod­ic move­ments. The last excerpt (2 rep­e­ti­tions) will prove — if at all nec­es­sary — that the into­na­tion of Indian ragas is much more than a sequence of notes in a scale, what­ev­er its accuracy:

S --> AsadAliKhan3
AsadAliKhan3 --> scale(todi_aak_3,0) 94/100 {26/10,sa3}{23/10,sa3 rek3 gak3}{195/100,ma#3}{111/100,rek3}{24/10,rek3 sa3}{33/10,sa3 sa3}{71/100,rek3}{76/100,gak3}{71/100,dhak3 ma#3}{176/100,dhak3}{75/100,sa4}{27/10,dhak3__ sa4}{620/100,sa4 dhak3 ma#3 dhak3 ma#3 gak3 _ ma#3 dhak3 dhak3&}{266/100,&dhak3}{672/100,pa3____ pa3_ pa3 pa3 pa3__}{210/100,pa3 ma#3 pa3 dhak3}{222/100,dhak3}{163/100,gak3 ma#3}{426/100,gak3_ rek3____}{346/100,sa3}

This melod­ic phrase is repeat­ed 2 times to check its super­im­po­si­tion with the sequence of notes pro­duced by the gram­mar
➡ This is a stereo record­ing. Use ear­phones to hear the music and sequence of plucked notes separately

Listen to Asad Ali Khan’s actu­al per­for­mance of raga Todi to appre­ci­ate its expres­sive power!

Trying to fol­low the intri­ca­cy of alankara (note treat­ment) with a sim­plis­tic nota­tion of melod­ic phras­es shows the dis­rup­tion between “model-based” exper­i­men­tal musi­col­o­gy and the real­i­ty of musi­cal prac­tice. This explains why we resort­ed to descrip­tive mod­els (e.g. auto­mat­ic nota­tion) cap­tured by the Melodic Movement Analyzer or com­put­er tools such as Praat, rather than attempt­ing to recon­struct melod­ic phras­es from the­o­ret­i­cal mod­els. Experiments on scales deal with the “skele­tal” nature of into­na­tion, which is a nec­es­sary yet not suf­fi­cient para­me­ter for describ­ing melod­ic types.

All exam­ples shown on this page are avail­able in the sam­ple set bp3-ctests-main.zip shared on GitHub. Follow instruc­tions on Bol Processor ‘BP3’ and its PHP inter­face to install BP3 and learn its basic oper­a­tion. Download and install Csound from its dis­tri­b­u­tion page.

Bernard Bel — Dec. 2020


References

Arnold, E.J.; Bel, B. L’intonation juste dans la théorie anci­enne de l’Inde : ses appli­ca­tions aux musiques modale et har­monique. Revue de musi­colo­gie, JSTOR, 1985, 71e (1-2), p.11-38.

Arnold, E.J. A Mathematical mod­el of the Shruti-Swara-Grama-Murcchana-Jati System. Journal of the Sangit Natak Akademi, New Delhi 1982.

Arnold, E.J.; Bel, B. A Scientific Study of North Indian Music. NCPA Quarterly Journal, vol. XII Nos. 2 3, Bombay 1983.

Arnold, W.J. Playing with Intonation. ISTAR Newsletter Nr. 3-4, June 1985 p. 60-62.

Bel, B. Musical Acoustics: Beyond Levy’s “Intonation of Indian Music”. ISTAR Newsletter Nr 2, April 1984.

Bel, B. A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra. Note interne, Groupe Représentation et Traitement des Connaissances (CNRS), March 1988a.

Bel, B. Raga : approches con­ceptuelles et expéri­men­tales. Actes du col­loque “Structures Musicales et Assistance Informatique”, Marseille 1988b.

Bel, B.; Bor, J. Intonation of North Indian Classical Music: work­ing with the MMA. National Center for the Performing Arts. Video on Dailymotion, Mumbai 1984.

Bharata. Natya Shastra. There is no cur­rent­ly avail­able English trans­la­tion of the first six chap­ters of Bharata’s Natya Shastra. However, most of the infor­ma­tion required for this inter­pre­ta­tion has been repro­duced and com­ment­ed by Śārṅgadeva in his Sangita Ratnakara (13th cen­tu­ry AD).

Bor, J.; Rao, S.; van der Meer, W.; Harvey, J. The Raga Guide. Nimbus Records & Rotterdam Conservatory of Music, 1999. (Book and CDs)

Bose, N.D. Melodic Types of Hindustan. Bombay, 1960: Jaico.

Rao, S.; Van der Meer, W. The Construction, Reconstruction, and Deconstruction of Shruti. Hindustani music: thir­teenth to twen­ti­eth cen­turies (J. Bor). New Delhi, 2010: Manohar.

Shringy, R.K.; Sharma, P.L. Sangita Ratnakara of Sarngadeva: text and trans­la­tion, vol. 1, 5: 7-9. Banaras, 1978: Motilal Banarsidass. Source in the Web Archive.

Van der Meer, W.; Rao, S. Microtonality in Indian Music: Myth or Reality. Gwalior, 2009: FRSM.

Van der Meer, W. Gandhara in Darbari Kanada, The Mother of All Shrutis. Pre-print, 2019.

A Mathematical Discussion of the Ancient Theory of Scales according to Natyashastra

Bel, B. A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra. Note interne, Groupe Représentation et Traitement des Connaissances (CNRS), Marseille 1988.

A Mathematical Model of the Shruti-Swara-Grama-Murcchana-Jati System

This is a scan of Arnold, E.J. A Mathematical mod­el of the Shruti-Swara-Grama-Murcchana-Jati System. Journal of the Sangit Natak Akademi, New Delhi 1982. This paper is quot­ed in The Two-Vina Experiment.

Melodic types of Hindustan

This a a scan of Bose, N.D. Melodic Types of Hindustan. Jaico, Bombay 1960. This paper is quot­ed in The Two-Vina Experiment.

Musicology

Microtonality
On elec­tron­ic instru­ments such as the Bol Processor asso­ci­at­ed with Csound, micro­tonal­i­ty is the mat­ter of “micro­ton­al tun­ing”, here mean­ing the con­struc­tion of musi­cal scales out­side the con­ven­tion­al one(s) …
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Just intonation: a general framework
A frame­work for con­struct­ing scales (tun­ing sys­tems) refer­ring to just into­na­tion in both clas­si­cal Indian and Western approach­es …
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The two-vina experiment
A com­pre­hen­sive inter­pre­ta­tion of the exper­i­ment of the two vinas described in Chapter XXVIII.24 of the Natya Shastra …
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Melodic types of Hindustan
A scan of Bose, N.D. Melodic Types of Hindustan. Jaico, Bombay 1960 …
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A Mathematical Model of the Shruti-Swara-Grama-Murcchana-Jati System
A scan of Arnold, E.J. A Mathematical mod­el of the Shruti-Swara-Grama-Murcchana-Jati System …
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A Mathematical Discussion of the Ancient Theory of Scales according to Natyashastra
Bel, B. A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra. Note interne, Groupe Représentation et Traitement …
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Raga intonation
This arti­cle demon­strates the the­o­ret­i­cal and prac­ti­cal con­struc­tion of micro­ton­al scales for the into­na­tion of North Indian ragas …
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Creation of just-intonation scales
The fol­low­ing is the pro­ce­dure for export­ing just-intonation scales from mur­ccha­na-s of Ma-grama stored in “-cs.12_scales”. ➡ Read Just into­na­tion: …
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A multicultural model of consonance
A frame­work for tun­ing just-intonation scales via two series of fifths For more than twen­ty cen­turies, musi­cians, instru­ment mak­ers and …
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Comparing temperaments
Images of tem­pered scales cre­at­ed by the Bol Processor The fol­low­ing are Bol Processor + Csound inter­pre­ta­tions of Bach’s Prelude …
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The two-vina experiment

The first six chap­ters of Natya Shastra, a Sanskrit trea­tise on music, dance and dra­ma dat­ing back to a peri­od between 400 BCE and 200 CE, con­tain the premis­es of a scale the­o­ry which for long caught the atten­tion of schol­ars in India and Western coun­tries. Early inter­pre­ta­tions by Western musi­col­o­gists fol­lowed the “dis­cov­ery” of this text in 1794 by philol­o­gist William Jones. Hermann Helmholtz’s the­o­ry of “nat­ur­al con­so­nance” gave way to many com­par­a­tive spec­u­la­tions based on phe­nom­e­na which Indian authors had ear­li­er observed as inher­ent to the “self-production” (svayamb­hū) of musi­cal notes (Iyengar 2017 p. 8).

Suvarnalata Rao and Wim van der Meer (2009) pub­lished a detailed record of attempts to elu­ci­date the ancien the­o­ry of musi­cal scales in the musi­co­log­i­cal lit­er­a­ture, com­ing back to the notions of ṣru­ti and swara which changed over time up to present-day musi­cal practice.

Accurate set­tings of the Shruti Harmonium (1980)

In the sec­ond part of the 20th cen­tu­ry, exper­i­men­tal work with fre­quen­cy meters led to con­tra­dic­to­ry con­clu­sions drawn from the analy­sis of small sam­ples of music per­for­mance. It was only after 1981 that sys­tem­at­ic exper­i­ments were con­duct­ed in India by the ISTAR team (E.J. Arnold, B. Bel, J. Bor and W. van der Meer) with an elec­tron­ic pro­gram­ma­ble har­mo­ni­um (the Shruti Harmonium) and lat­er a “micro­scope” for melod­ic music, the Melodic Movement Analyser (MMA) (Arnold & Bel 1983, Bel & Bor I985) feed­ing accu­rate pitch data to a com­put­er to process hours of music select­ed from his­tor­i­cal recordings.

After sev­er­al years of exper­i­men­tal work, it had become clear that, even though the into­na­tion of Indian clas­si­cal music is far from a ran­dom process, it would be haz­ardous to assess an inter­pre­ta­tion of ancient scale the­o­ry with the aid of today’s musi­cal data. There at least three rea­sons for this:

  1. There are infi­nite­ly valid inter­pre­ta­tions of the ancient the­o­ry, as we will show.
  2. The con­cept of raga, i.e. the basic prin­ci­ple of Indian clas­si­cal music, appeared first in the lit­er­a­ture cir­ca 900 CE in Matanga’s Brihaddeshi, and it under­went a grad­ual devel­op­ment until 13th cen­tu­ry, when Sharangadeva enlist­ed 264 ragas in his Sangitratnakara.
  3. Drones were not in use at the time of Natya Shastra; the influ­ence of the drone on into­na­tion is con­sid­er­able, if not pre­dom­i­nant, in con­tem­po­rary music performance.

The ancient Indian the­o­ry of scales remains use­ful for its insight into ear­ly melod­ic clas­si­fi­ca­tion (the jāti sys­tem) which lat­er might have engen­dered the raga sys­tem. Therefore, it may be best envis­aged as a topo­log­i­cal descrip­tion of tonal struc­tures. Read Raga into­na­tion for a more detailed account of the­o­ret­i­cal and prac­ti­cal issues.

The top­ic of this page is an inter­pre­ta­tion of the exper­i­ment of the two vinas described in Chapter XXVIII.24 of the Natya Shastra. An analy­sis of the under­ly­ing mod­el has been pub­lished in A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra (Bel 1988) which the fol­low­ing pre­sen­ta­tion will ren­der more comprehensive.

The historical context

Bharata Muni, the author(s) of Natya Shastra must have heard about the­o­ries of musi­cal scales attrib­uted to ancient Greeks. Indian schol­ars were in posi­tion to bor­row the mod­el and expand it con­sid­er­ably because of their deep under­stand­ing of mathematics.

Readers of C.K. Raju — notably his out­stand­ing work Cultural Foundations of Mathematics (2007) — are aware that Indian mathematicians/philosophers are not only famous for invent­ing posi­tion­al nota­tion which took six cen­turies to be adopt­ed in Europe… They also laid out the foun­da­tions of cal­cu­lus and infin­i­tes­i­mals, lat­er export­ed by Jesuit priests from Kerala to Europe and borrowed/appropriated by European schol­ars (Raju 2007 pages 321-373).

The cal­cu­lus first devel­oped in India as a sophis­ti­cat­ed tech­nique to cal­cu­late pre­cise trigono­met­ric val­ues need­ed for astro­nom­i­cal mod­els. These val­ues were pre­cise to the 9th place after the dec­i­mal point; this pre­ci­sion was need­ed for the cal­en­dar, crit­i­cal to monsoon-driven Indian agri­cul­ture […]. This cal­cu­la­tion involved infi­nite series which were summed using a sophis­ti­cat­ed phi­los­o­phy of ratios of inex­pressed num­bers [today called ratio­nal functions…].

Europeans, how­ev­er, were prim­i­tive and back­ward in arith­meti­cal cal­cu­la­tions […] and bare­ly able to do finite sums. The dec­i­mal sys­tem had been intro­duced in Europe by Simon Stevin only at the end of the 16th c., while it was in use in India since Vedic times, thou­sands of years earlier.

C. K. Raju (2013 p. 161- 162)

This may be cit­ed in con­trast with the state­ments of Western his­to­ri­ans, among which:

The his­to­ry of math­e­mat­ics can­not with cer­tain­ty be traced back to any school or peri­od before that of the Greeks […] though all ear­ly races knew some­thing of numer­a­tion […] and though the major­i­ty were also acquaint­ed with the ele­ments of land-surveying, yet the rules which they pos­sessed […] were nei­ther deduced from nor did they form part of any science.

W. W. Rouse Ball, A Short Account of the History of Mathematics. Dover, New York, 1960, p. 1–2.

Therefore, it may seem para­dox­i­cal, giv­en such an intel­lec­tu­al bag­gage, to write an entire chap­ter on musi­cal scales with­out a sin­gle num­ber! In A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra I showed a min­i­mal rea­son: Bharata’s descrip­tion leads to an infi­nite set of solu­tions that should be for­mal­ized with alge­bra rather than a finite set of numbers.

The experiment

The author(s) of Natya Shastra invite(s) the read­er to take two vina-s (plucked stringed instru­ments) and tune them on the same scale.

A word of cau­tion to clar­i­fy the con­text: this chap­ter of Natya Shastra may be read as a thought exper­i­ment rather than a process involv­ing phys­i­cal objects. There is no cer­ti­tude that these two vina-s ever exist­ed — and even that “Bharata Muni”, the author/experimenter, was a unique per­son. Their approach is an instance of a val­i­da­tion (pramāņa) resort­ing to empir­i­cal proof, in oth­er words dri­ven by the phys­i­cal­ly man­i­fest (pratyakşa) rather than inferred from “axioms” con­sti­tu­tive of a the­o­ret­i­cal mod­el. This may be summed up as “pre­fer­ring physics to metaphysics”.

Constructing and manip­u­lat­ing vina-s in the man­ner indi­cat­ed by the exper­i­menter seems an insur­montable tech­no­log­i­cal chal­lenge. This has been a sub­ject of dis­cus­sion among a num­ber of authors — read Iyengar (2017 pages 7-sq.) Leaving aside the pos­si­bil­i­ty of a prac­ti­cal real­iza­tion is not a denial of phys­i­cal real­i­ty, as for­mal math­e­mat­ics would sys­tem­at­i­cal­ly man­date. Calling it a “thought exper­i­ment” is a way of assert­ing the link with the phys­i­cal mod­el. In the same man­ner, using cir­cu­lar graphs to rep­re­sent tun­ing schemes and alge­bra to describe rela­tions between inter­vals are aids to under­stand­ing which do not reduce the mod­el to spe­cif­ic, ide­al­is­tic inter­pre­ta­tions sim­i­lar to spec­u­la­tions on inte­ger num­bers cher­ished by Western schol­ars. These graphs aim at facil­i­tat­ing a com­pu­ta­tion­al design of instru­ments mod­el­ing these imag­ined instru­ments — read Raga into­na­tion and Just into­na­tion, a gen­er­al frame­work.

Let us fol­low Bharata’s instruc­tions and tune both instru­ments on a scale called “Sa-grama” about which the author declares:

The sev­en notes [svaras] are: Şaḍja [Sa], Ṛşbha [Ri], Gāndhāra [Ga], Madhyama [Ma], Pañcama [Pa], Dhaivata [Dha], and Nişāda [Ni].

It is tempt­ing to iden­ti­fy this scale as the con­ven­tion­al Western seven-grade scale do, re, mi, fa, sol, la, si (“C”, “D”, “E”, “F”, “A”, “B”) which some schol­ars have done despite the faulty inter­pre­ta­tion of intervals.

Intervals are notat­ed in shru­ti-s which may, for a start, be tak­en as an order­ing sys­tem rather than a unit of mea­sure­ment. The exper­i­ment will con­firm that a four-shru­ti inter­val is larg­er than a three-shru­ti, a three-shru­ti larg­er than two-shru­ti and the lat­ter larg­er than a sin­gle shru­ti. In dif­fer­ent con­texts the word “shru­ti” des­ig­nates note posi­tions instead of inter­vals between notes. This ambi­gu­i­ty is also a source of confusion.

The author writes:

Śrutis in the Şaḍja Grāma are shown as fol­lows: three [in Ri], two [in Ga], four [in Ma], four [in Pa], three [in Dha], two [in Ni], and four [in Sa].

Bharata enlists 9-shru­ti (con­so­nant) inter­vals: “Sa-Pa”, “Sa-Ma”, “Ma-Ni”, “Ni-Ga” and “Re-Dha”. In addi­tion, he defines anoth­er scale named “Ma-grama” in which “Pa” is one shru­ti low­er than “Pa” in the Sa-grama, so that “Sa-Pa” is no longer con­so­nant where­as “Re-Pa” is con­so­nant because it is made of 9 shru­ti-s.

Intervals of 9 or 13 shru­ti-s are declared “con­so­nant” (sam­va­di). Leaving out the octave, the best con­so­nance in a musi­cal scale is the per­fect fifth with a fre­quen­cy ratio close to 3/2. When tun­ing stringed instru­ments, a ratio dif­fer­ing from 3/2 gen­er­ates beats indi­cat­ing that a string is out of tune.

Sa-grama and Ma-grama accord­ing to Natya Shastra. Red and green seg­ments indi­cate the two chains of per­fect fifths. Underlined note names denote ‘flat’ positions.

If fre­quen­cy ratios are expressed log­a­rith­mi­cal­ly with 1200 cents rep­re­sent­ing one octave, and fur­ther con­vert­ed to angles with a full octave on a cir­cle, the descrip­tion of Sa-grama and Ma-grama scales may be sum­ma­rized in a cir­cu­lar dia­gram (see picture).

Two cycles of fifths have been high­light­ed in red and green col­ors. Note that both the “Sa-Ma” and “Ma-Ni” inter­vals are per­fect fifths, dis­card­ing the asso­ci­a­tion of Sa-grama with the con­ven­tion­al Western scale: the “Ni” should be mapped to “B flat”, not to “B”. Further, the “Ni-Ga” per­fect fifth implies that “Ga” is also “E flat” rather than “E”. The Sa-grama and Ma-grama scales are there­fore “D modes”. For this rea­son, “Ga” and “Ni” appear under­lined on the diagrams.

Authors eager to iden­ti­fy Sa-grama and Ma-grama as a Western scale claimed that when the text says that there are “3 shruti-s in Re” it should be under­stood between Re and Ga. Yet this inter­pre­ta­tion is incon­sis­tent with the sec­ond low­er­ing of the move­able vina (see below).

We must avoid pre­ma­ture con­clu­sions about inter­vals in these scales. The two cycles of fifths are unre­lat­ed except that the “dis­tance” between the “Pa” of Ma-grama and that of Sa-grama is “one-shru­ti”:

The dif­fer­ence which occurs in Pañcama when it is raised or low­ered by a Śruti and when con­se­quen­tial slack­ness or tense­ness [of strings] occurs, will indi­cate a typ­i­cal (pramāņa) Śruti. (XXVIII, 24)

In oth­er words, the size of this pramāņa ṣru­ti is not spec­i­fied. It would there­fore be mis­lead­ing to pos­tu­late its equiv­a­lence with the syn­ton­ic com­ma (fre­quen­cy ratio 81/80). Doing so reduces Bharata’s mod­el to “just into­na­tion”, indeed with inter­est­ing prop­er­ties in its appli­ca­tion to Western har­mo­ny (read page), but with ques­tion­able rel­e­vance to the prac­tice of Indian music. As claimed by Arnold (1983 p. 39):

The real phe­nom­e­non of into­na­tion in Hindustani Classical Music as prac­tised is much more amor­phous and untidy than any geom­e­try of course, as recent empir­i­cal stud­ies by Levy (1982), and Arnold and Bel (1983) show.

The des­ig­na­tion of the small­est inter­val as “pramāņa ṣru­ti ” is of major epis­temic rel­e­vance and deserves a brief expla­na­tion. The seman­tics of “slack­ness or tense­ness” clear­ly belongs to “pratyakṣa pramāṇa”, the means of acquir­ing knowl­edge by per­cep­tu­al expe­ri­ence. More pre­cise­ly, “pramāṇa (प्रमाण) refers to “valid per­cep­tion, mea­sure and struc­ture”” (Wisdom Library), a notion of proof shared by all Indian tra­di­tion­al schools of phi­los­o­phy (Raju 2007 page 63). We will get back to this notion in the conclusion.

An equiv­a­lent way of con­nect­ing the two cycles of fifths would be to define a 7-shru­ti inter­val, for instance “Ni-Re”. If the pramāņa ṣru­ti were a syn­ton­ic com­ma then this inter­val would be a har­mon­ic major third with ratio 5/4. As evoked in Just into­na­tion, a gen­er­al frame­work, the inven­tion of the major third as a con­so­nant inter­val dates back to the ear­ly 16th cen­tu­ry in Europe. In Natya Shastra this 7-shru­ti inter­val had been rat­ed “asso­nant” (anu­va­di).

In all writ­ings refer­ring to the ancient Indian the­o­ry of scales, I occa­sion­al­ly used “pramāņa ṣru­ti” and “syn­ton­ic com­ma” as equiv­a­lent terms. This is accept­able if one keeps in mind that the syn­ton­ic com­ma is allowed to take val­ues oth­er than 81/80. Consequently, the “har­mon­ic major third” should not auto­mat­i­cal­ly be assigned fre­quen­cy ratio 5/4.

Picture above rep­re­sents the two vina-s tuned iden­ti­cal­ly on Sa-grama. Matching notes are marked by yel­low spots. The inner part of the blue cir­cle will be the mov­able vina in the fol­low­ing trans­po­si­tion process­es, and the out­er part the fixed vina.

First lowering

Bharata writes:

The two Vīņās with beams (danḍa) and strings of sim­i­lar mea­sure, and with sim­i­lar adjust­ment of the lat­ter in the Şaḍja Grāma should be made [ready]. [Then] one of these should be tuned in the Madhyama Grāma by low­er­ing Pañcama [by one Śruti]. The same (Vīņā) by adding one Śruti (lit. due to the adding of one Śruti) to Pañcama will be tuned in the Şaḍja Grāma.

In brief, this is a pro­ce­dure for low­er­ing all notes of the mov­able vina by one pramāņa ṣru­ti. First low­er its “Pa” — e.g. make it con­so­nant with “Re” of the fixed vina — to obtain Ma-grama on the mov­able vina. Then read­just its whole scale to obtain Sa-grama. Note that low­er­ing “Re” and “Dha” implies appre­ci­at­ing again the size of a pramāņa ṣru­ti while pre­serv­ing the “Re-Dha” con­so­nant inter­val. The result is as follows:

The two vinas after a low­er­ing of pramāņa ṣru­ti

The pic­ture illus­trates the fact that there are no more match­ing notes between the two vina-s.

Interpreting shruti-s as vari­ables in some metrics

This sit­u­a­tion can be trans­lat­ed to alge­bra. Let “a”, “b”, “c” … “v” be the unknown sizes of shru­ti-s in the scale (see pic­ture on the side). A met­rics “trans­lat­ing” Bharata’s mod­el will be nec­es­sary for check­ing it on sound struc­tures pro­duced by an elec­tron­ic instru­ment — the com­put­er. The scope of this trans­la­tion remains valid as long as no extra asser­tion has been stat­ed which is not root­ed in the orig­i­nal model.

Using sym­bol “#>” to indi­cate that two notes are not match­ing, this first low­er­ing may be sum­ma­rized by the fol­low­ing set of inequations:

s + t + u + v > m 
a + b + c > m 
d + e > m 
f + g + h + i > m 
n + o + p > m 
q + r > m 
Sa #> Ni
Re #> Sa
Ga #> Re
Ma #> Ga
Dha #> Pa
Ni #> Dha

Second lowering

The next step is again a low­er­ing by one shru­ti with a dif­fer­ent procedure.

Again due to the decrease of a Śruti in anoth­er [Vīņā], Gāndhāra and Nişāda will merge with Dhaivata and Ṛşbha respec­tive­ly, when there is an inter­val of two Śrutis between them.

Note that it is no longer pos­si­ble to rely on a low­ered “Pa” to eval­u­ate a pramāņa ṣru­ti for the low­er­ing. The instruc­tion is to low­er the tun­ing of the mov­able vina until either “Re” and “Ga” or “Dha” and “Ni” are merged, which is claimed to be the same because of the final low­er­ing of two shru­ti-s (from the ini­tial state):

The two vina-s after the sec­ond low­er­ing (2 shru­ti-s)

Now we get an equa­tion report­ing that the two-shru­ti inter­vals are equal in size:

q + r = d + e

and five more inequa­tions indi­cat­ing the non-matching of oth­er notes:

f + g + h + i > d + e
a + b + c > d + e
s + t + u + v > d + e
n + o + p > d + e
j + k + l + m > d + e
Ma #> Ga
Re #> Sa
Sa #> Ni
Dha #> Pa
Pa #> Ma

We should keep in mind that the author is describ­ing a phys­i­cal process, not an abstract “move­ment” by which the move­able wheel (or vina) would “jump in space” from its ini­tial to final posi­tion. Therefore we pay atten­tion to things hap­pen­ing and not hap­pen­ing dur­ing the tun­ing of the vina, or rota­tion of the wheel, look­ing at the tra­jec­to­ries of dots rep­re­sent­ing note posi­tions (along the blue cir­cle). Things not hap­pen­ing (non-matching notes) yield inequa­tions required for mak­ing sense of the alge­bra­ic model.

This step of the exper­i­ment con­firms that it is wrong to locate Sa at the posi­tion of Ni for the sake of iden­ti­fy­ing Sa-grama with the Western scale. In this case, match­ing notes would no be Re-Ga and Dha-Ni, but Ga-Ma and Ni-Sa.

Third lowering

Bharata writes:

Again due to the decrease of a Śruti in anoth­er [Vīņā], Ṛşbha and Dhaivata will merge with Şaḍja and Pañcama respec­tive­ly, when there is an inter­val of three Śrutis between them.

The two vinas after the third low­er­ing (3 shruti-s)

This leads to equation

n + o + p = a + b + c

and inequa­tions:

s + t + u + v > a + b + c
f + g + h + i > a + b + c
j + k + l + m > a + b + c
Sa #> Ni
Ma #> Ga
Pa #> Ma

Fourth lowering

The pro­ce­dure:

Similarly the same [one] Śruti being again decreased, Pañcama, Madhyama and Şaḍja will merge with Madhyama, Gāndhāra and Nişāda respec­tive­ly when there is an inter­val of four Śrutis between them.

The two vinas after the fourth low­er­ing (4 shruti-s)

This yields 2 equations:

j + k + l + m = f + g + h + i
s + t + u + v = f + g + h + i

Algebraic interpretation

After elim­i­nat­ing redun­dant equa­tions and inequa­tions, con­straints are sum­ma­rized as follows:

(S1) d + e > m
(S2) a + b + c > d + e
(S3) f + g + h + i > a + b + c
(S4) j + k + l + m = f + g + h + i
(S5) s + t + u + v = f + g + h + i
(S6) n + o + p = a + b + c
(S7) q + r = d + e

The three inequa­tions illus­trate the fact that num­bers of shru­ti-s denote an order­ing of the sizes of inter­vals between notes.

Still, we have 22 unknown vari­ables and only 4 equa­tions. These vari­ables can be “packed” to a set of 8 unknown vari­ables which rep­re­sent the “macro-intervals”, i.e. the steps of the gra­ma-s. In this approach, shru­ti-s are sort of “sub­atom­ic” par­ti­cles which these “macro-intervals” are made of… Now we need only 4 aux­il­iary equa­tions to deter­mine the scale. These may be pro­vid­ed by acoustic infor­ma­tion, with inter­vals mea­sured in cents. First we express that the sum of the vari­ables, the octave, is equal to 1200 cents: 

(S8) (a + b + c) + (d + e) + (f + g + h + i) + (j + k + l) + m + (n + o + p) + (q + r) + (s + t + u + v) = 1200

Then we inter­pret all sam­va­di rela­tion­ships as per­fect fifths (ratio 3/2 = 701.9 cents):

(S9) (a + b + c) + (d + e) + (f + g + h + i) + (j + k + l) + m = 701.9 (Sa-Pa)
(S10) (j + k + l) + m + (n + o + p) + (q + r) + (s + t + u + v) = 701.9 (Ma-Sa)
(S11) (d + e) + (f + g + h + i) + (j + k + l) + m + (n + o + p) = 701.9 (Re-Dha)
(S12) (f + g + h + i) + (j + k + l) + m + (n + o + p) + (q + r) = 701.9 (Ga-Ni)

includ­ing the “Re-Pa” per­fect fifth in Ma-grama:

(S13) m + (n + o + p) + (q + r) + (s + t + u + v) + (a + b + c) = 701.9

S1O, S11 and S12 can all be derived from S9. These equa­tions may there­fore be dis­card­ed. We still need one more equa­tion to solve the sys­tem. At this stage there are many options asso­ci­at­ed with tun­ing pro­ce­dures. As sug­gest­ed above, set­ting the har­mon­ic major third to ratio 5/4 (386.3 cents) would pro­vide the miss­ing equa­tion. This amounts to set­ting vari­able “m” to 21.4 cents (syn­ton­ic com­ma). However, this major third can take any val­ue up to the Pythagorean third (81/64 = 407.8 cents) for which we would get m = 0.

Beyond this range, the two-vina exper­i­ment is no longer valid, but it leaves a great amount of pos­si­bil­i­ties includ­ing the tem­pera­ment of some inter­vals which musi­cians might achieve spon­ta­neous­ly in par­al­lel melod­ic move­ments. A set of solu­tions is exposed in A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra and a few of them have been tried on the Bol Processor to check musi­cal exam­ples for which they might pro­vide ade­quate scales — read Raga into­na­tion.

Extensions of the model

In order to com­plete his sys­tem of scales, Bharata had to intro­duce two new notes in the basic gra­ma-s: antara Gandhara and kakali Nishada. The new “Ga” is defined as “G” raised by 2 shru­ti-s. Similarly, kakali Ni is “N” raised by 2 shru­ti-s.

In order to posi­tion “Ni” and “Ga” cor­rect­ly we must inves­ti­gate the behav­ior of the new scale in all trans­po­si­tions (mur­ccha­na-s), includ­ing those start­ing with “Ga” and “Ni”, and infer equa­tions cor­re­spond­ing to an opti­mal con­so­nance of the scale. We end up with 11 equa­tions for only 10 unknown vari­ables, which means that this per­fec­tion can­not be achieved. One con­straint must be released.

An option is to release con­straints on major thirds, fifths or octaves, lead­ing to a form of tem­pera­ment. For instance, stretch­ing the octave by 3.7 cents gen­er­ates per­fect fifths (701.9 cents) and har­mon­ic major thirds close to equal tem­pera­ment (401 cents) with a com­ma of 0 cents. This tun­ing tech­nique was advo­cat­ed by Serge Cordier (Asselin 2000 p. 23; Wikipedia).

An equal-tempered scale with octave stretched at 1204 cents. (Image cre­at­ed by Bol Processor BP3)

Another option is to come as close as pos­si­ble to “just into­na­tion” with­out mod­i­fy­ing per­fect fifths and octaves. This is pos­si­ble if the com­ma (vari­able “m”) is allowed an arbi­trary val­ue between 0 and 56.8 cents. Limits are imposed by the inequa­tions derived from the two-vina experiment.

These “just sys­tems” are cal­cu­lat­ed as follows:

a + b + c = j + k + l = n + o + p = Maj - C
d + e = h + i = q + r = u + v = L + C
f + g = s + t = Maj - L - C
m = C

where L = 90.25 cents (lim­ma = 256/243), Maj = 203.9 cents (major who­le­tone = 9/8)
and 0 < C < 56.8 (pramāņa ṣru­ti or syn­ton­ic comma)

This leads to the 53-grade scale named “gra­ma” which we use as a frame­work for con­so­nant chro­mat­ic scales eli­gi­ble for pure into­na­tion in Western har­mo­ny when the syn­ton­ic com­ma is sized 81/80. Read Just into­na­tion, a gen­er­al frame­work:

The “gra­ma” scale used for just into­na­tion, with a syn­ton­ic com­ma of 81/80. Pythagorean cycle of fifths in red, har­mon­ic cycle of fifths in green.

In BP3, the just-intonation frame­work has been extend­ed so that any val­ue of the syn­ton­ic com­ma (or the har­mon­ic major third) can be set on a giv­en scale struc­ture. This fea­ture is demon­strat­ed on page Raga into­na­tion.

The relevance of circular representations


Circular rep­re­sen­ta­tion of tāl Pañjābi, catuśra­jāti
[16 counts] from a Gujarati text in Devanagari script
(J. Kippen, pers. communication)

It is safe to clas­si­fy the two-vina exper­i­ment as thought exper­i­ment because of the unlike­li­hood that it could be worked with mechan­i­cal instru­ments. Representing it on a cir­cu­lar graph (a move­able wheel inside a fixed crown) achieves the same goal with­out resort­ing to imag­i­nary devices.

Circular rep­re­sen­ta­tions belong to Indian tra­di­tions of var­i­ous schools, among which the descrip­tion of rhyth­mic cycles (tāl-s) used by drum play­ers. These graphs are meant to out­line the rich inter­nal struc­ture of musi­cal con­struc­tions that can­not be reduced to “beat count­ing” (Kippen 2020).

For instance, the image above was used to describe the ţhekkā (cycle of quasi-onomatopoeic syl­la­bles rep­re­sent­ing the drum strokes) of tāl Pañjābi which reads as follows:

Unfortunately, ear­ly print­ing press tech­nol­o­gy may have ren­dered uneasy the pub­li­ca­tion and trans­mis­sion of these aids to learning.

If con­tem­po­raries of Bharata ever used sim­i­lar cir­cu­lar rep­re­sen­ta­tions for reflect­ing on musi­cal scales, we guess that arche­o­log­i­cal traces might not be iden­ti­fied prop­er­ly as their draw­ings could be mis­tak­en for yantra-s, astro­log­i­cal charts and the like!

Return to epistemology

Bharata’s exper­i­ment is a typ­i­cal exam­ple of the pref­er­ence for facts inferred from empir­i­cal obser­va­tions over a pro­claimed uni­ver­sal log­ic aimed at estab­lish­ing “irrefragable demonstrations”.

Empirical proofs are uni­ver­sal, not meta­phys­i­cal proofs; elim­i­nat­ing empir­i­cal proofs is con­trary to all sys­tems of Indian phi­los­o­phy. Thus ele­vat­ing meta­phys­i­cal proofs above empir­i­cal proofs, as for­mal math­e­mat­ics does, is a demand to reject all Indian phi­los­o­phy as infe­ri­or. Curiously, like Indian phi­los­o­phy, present-day sci­ence too uses empir­i­cal means of proof, so this is also a demand to reject sci­ence as infe­ri­or (to Christian metaphysics).

Logic is not uni­ver­sal either as Western philoso­phers have fool­ish­ly main­tained: Buddhist [qua­si truth-functional] and Jain [three-valued] log­ics are dif­fer­ent from those cur­rent­ly used in for­mal math­e­mat­i­cal proof. The the­o­rems of math­e­mat­ics would change if those log­ics were used. So, impos­ing a par­tic­u­lar log­ic is a means of cul­tur­al hege­mo­ny. If log­ic is decid­ed empir­i­cal­ly, that would, of course, kill the phi­los­o­phy of meta­phys­i­cal proof. Further, it may result in quan­tum log­ic, sim­i­lar to Buddhist logic […].

C. K. Raju (2013 p. 182-183)
Yuktibhāşā’s proof of the “Pythagorean” the­o­rem.
Source: C. K. Raju (2007 p. 67)

The two-vina exper­i­ment can be likened to the (more recent) phys­i­cal proof of the “Pythagorean the­o­rem”. This the­o­rem (Casey 1885 p. 43) was known in India and Mesopotamia long before the time of its leg­endary author (Buckert 1972 p. 429, 462). In the Indian text Yuktibhāşā (ca. 1530 CE), a fig­ure of a right-angle tri­an­gle is drawn on a palm leaf with squares on its two sides and its hypothenuse. Then the fig­ure is cut and rotat­ed in a way high­light­ing that the areas are equal.

Clearly, the proof of the “Pythagorean the­o­rem” is very easy if one is either (a) allowed to make mea­sure­ments, or, equiv­a­lent­ly (b) allowed to move fig­ures about in space.

C. K. Raju (2013 p. 167)

This process takes place on sev­er­al steps of mov­ing fig­ures in a way sim­i­lar to mov­ing scales (or fig­ures rep­re­sent­ing scales) in the two-vina exper­i­ment. The 3 single-shru­ti tonal inter­vals may be likened to the areas of the 3 squares in Yuktibhāşā. The fol­low­ing remark would there­fore apply to Bharata’s procedure:

The details of this ratio­nale are not our imme­di­ate con­cern beyond observ­ing that draw­ing a fig­ure, car­ry­ing out mea­sure­ments, cut­ting, and rota­tion are all empir­i­cal pro­ce­dures. Hence, such a demon­stra­tion would today be reject­ed as invalid sole­ly on the ground that it involves empir­i­cal pro­ce­dures that ought not to be any part of math­e­mat­i­cal proof.

C. K. Raju (2007 p. 67)

Bernard Bel — Dec. 2020

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